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Duration and 


Duration and Simultaneity 


Henri Bergson 

Translated by 
Leon Jacobson 
Professor of Art, East Carolina College 

With an Introduction by 
Herbert Dingle 
Professor Emeritus of History and Philosophy of Science 
University of London 

The Library of Liberal Arts 

published by 
A Subsidiary of Howard W. Sams & Co., Inc. 
Publishers • Indianapolis • New York • Kansas City 

Henri Bergson: 1859-1941 

Duree et Simultaneity was originally published in 1922 


Librar^r 1 ? Unked States of America 

y ot Congress Catalog Card Number 64-66064 

First Printing 

Translator's Preface 

It is the moral of Bergson's philosophy that we shall not live 
as fully as we could, and that philosophy and science will not 
co-operate as they might, as long as we remain unaware that 
"it is time which is happening and, more than that, which 
causes everything to happen." If we do not notice the actuality 
and efficacy of time, it is not through oversight, but because 
time is ruled out by the intelligence, whether exercised in our 
daily problem-solving or, much more precisely, in scientific 
investigation. For our intellect was made to prepare our action 
upon things; and action is taken on fixed points. Our intelli- 
gence, looking for fixity, masks the flow of time by conceiving 
it as a juxtaposition of "instants" on a line. 

But, in Bergson's view, despite this normal exteriorization 
of our feeling of duration into a "spatialized" time, the mind, 
being more than intellect, is still capable of apprehending uni- 
versal becoming in a vision in which "what was immobile and 
frozen in our perception is warmed and set in motion." It is 
possible to "reascend the slope of nature" and, by a concen- 
trated effort of attention, by "intuition," to contact directly, 
deep within, that concrete duration which is "the very stuff of 
our existence and of all things." 

Bergson well understood, then, that it is our practical rou- 
tine that has militated against a renewal, or deepening, of our 
perception; that "our senses and our consciousness have re- 
duced real time and real change to dust in order to facilitate 
our action upon things." Nor, certainly, does he condemn 
positive science for not being concerned with duration (even 
though that is its inspiration), since "the function of science 
is, after all, to compose a world for us in which we can, for the 
convenience of action, ignore the effects of time." What he 
deplores, however, is the tendency of science, and philosophy, 



to mistake its conceptualizations of reality for reality itself. 
It is, indeed, against a biological and psychological "meta- 
physics" that Bergson's major works are directed, always with 
the ultimate aim of clearing the path to vision. Duration and 
Simultaneity is the concluding chapter in this long polemic 
with scientism. 

In the work before us, Bergson argues against the demand 
by "the theoreticians of relativity," made in the name of Ein- 
stein's theory of special relativity, that we believe in the "slow- 
ing" of time by motion in each relatively moving system in 
the universe. Of course, the very notion of slowed times runs 
counter to the common-sense view of a single, universal time; 
and it also contradicts Bergson's allied conception of duration, 
central in his philosophy. It therefore becomes Bergson's pur- 
pose in Duration and Simultaneity to demonstrate: (1) that it 
is actually the supposition of multiple, real times, not that of 
a single, real time, which Einstein's postulate of the reciprocity 
of motion contradicts; (2) that the considering of Einstein's 
times as "real" is attributable to an oscillation, in the course 
of physical investigation, between the standpoints of Einstein's 
"bilateral" and Lorentz' "unilateral" theory of relativity; and 
(3) that this oscillation is itself traceable to "our not having 
first analyzed our representation of the time that flows, our 
feeling of real duration." Let us first consider this last, and 
widest, "frame" of Bergson's argument. 

As in all his works, Bergson points out in Duration and 
Simultaneity that it is not the experience of duration that we 
ordinarily have in mind when we speak of time, but its meas- 
urement. For what we care about in practical life is the meas- 
urement of the real and not its nature. But we cannot di- 
rectly measure that reality which is duration, since it is an 
indivisible flow, and therefore has no measurable parts. To 
be measured, it must first be spatialized. Now, the first step 
in this process is taken when we think of the experienced flow 
of our mner duration as motion in space; and the next, when 
we agree to consider the path described by this motion as 

translator's preface 


the motion itself. In dividing and measuring the path, we then 
say we are dividing and measuring the duration of the mo- 
tion that is tracing it. 

For us, it is the earth's rotation that is the model motion 
tracing the path of time. Time then seems to us "like the un- 
winding of a thread, like the journey of the mobile [the earth] 
entrusted with measuring it. We shall then say that we have , 
measured the time of this unwinding and of the universal 
unwinding as well." But, if we can correlate these two un- 
windings, it is only because we have at our disposal the con- 
cept of simultaneity; and we owe this concept to our ability 
to perceive external flows of events either together with the 
flow of our own duration, or separately from it, or, still better, 
both separately and together, at one and the same time. If 
we then refer to two external flows which take up the same 
duration as being "simultaneous," it is because they abide 
within the duration of yet a third, our own. But, to be 
useable, these simultaneities of durations must be converted 
into simultaneities of instants; and this we do as soon as we 
have learned to spatialize time. As noted above, we divide the 
path that has come to symbolize the flow of real time into 
equal units of space, and into "instants," which are the ex- 
tremities of these units. But, now, in addition, we point off 
the whole length of the moving path of a contemporary event 
with corresponding points of division. Any portion of the 
duration of its motion is then considered measured when we 
have counted a number of such correspondences, or simul- 

These simultaneities are instantaneities, not partaking of 
the real time that endures. But they are yet simultaneous with 
instants pointed off by them along our inner duration, and 
created in the very act of pointing. Bergson declares, there-: 
fore, that it is because instant-simultaneities are imbedded in 
flow-simultaneities, and because the latter are referrable to 
our own duration, that what we are measuring is time as: 
well as space; and, conversely, if the time being measured is 



not finally convertible into an experienced duration, it is not 
time, but space, which we are measuring. 

Now, it happens that none of the motion-induced slowings 
of time allegedly uncovered by Einstein's theory of special 
relativity is convertible into duration. For, from Einstein's 
standpoint of the reciprocity of motion in space, these times 
are merely attributed by a real physicist-observer in a con- 
ventionally stabilized, "referrer" system S, to merely imagined 
physicist-observers in a conventionally mobilized, "referred-to" 
system 5'. Not being "pasted" to a time which is either lived 
or livable, they are purely fictional, in no way comparable 
to the actually lived time of the physicist in S. 

But the unreality of multiple times betokens the singleness 
of real time. For, were the referrer-physicist in S to betake 
himself to S', he would, by that very fact, be immobilizing it 
into a referrer system and would then live the same time 
there which he had lived in the former referrer system S. This 
mterchangeability of observers and their lived times in two 
systems in a state of uniform and reciprocal motion is conse- 
quent upon Einstein's hypothesis of the reciprocity of motion 
in space. Hence, "far from ruling out the hypothesis of a 
single, universal duration, Einstein's theory of special rela- 
tivity calls for it and confers upon it a superior intelligibility." 

The fact is, according to Bergson, that it is in Lorentz' 
unilateral," not Einstein's "bilateral" theory of relativity 
that multiple times can logically be considered real. For, it is 
there alone that a system of reference is regarded as at abso- 
lute rest, while other systems are in absolute motion. These 
conditions, found in Lorentz' theory, do imply the existence 
ot multiple times, all on the same footing and all "real." Yet, 
physicists support Einstein's, not Lorentz' theory of relativity; 
the question arises as to why they should attribute to 
instein a doctrine properly ascribable to Lorentz. To 
«Jr n ,' confusi °n of Einstein's and Lorentz' viewpoints 
seems almost inevitable. It stems from the fact that even when 

P ysicist begins by granting Einstein's thesis that any two 

translator's preface 


systems, S and S', are in reciprocal motion, he cannot, as a 
physicist, investigate this system without immobilizing one of 
them into a "stationary" system of reference. The result is that 
"absolute rest expelled by the understanding is reinstated by 
the imagination." In the mind of the physicist, two representa- 
tions of relativity then accompany one another, one, "radical 
and conceptual" (Einstein's), and the other, "attenuated and 
imagist" (Lorentz'), and "the concept undergoes contamina- 
tion by the image." In other words, even though the physicist 
conceives relativity from the standpoint of Einstein, he sees it 
a little from that of Lorentz. The multiple times— as well as the 
contractions in length, and dislocations of simultaneity into 
succession— which occur upon the application of the Lorentz 
transformation equations to a "moving" system, then appear 
real, as much in Einstein's as in Lorentz' theory of relativity. 

This point is an essential part of Bergson's demonstration 
of the compatibility of his philosophy of duration with the 
considerations of time in Einstein's theory of relativity. This 
demonstration is, of course, Bergson's main objective in Dura- 
tion and Simultaneity. But now, another and more general 
question arises as to how physicists have been led, in the first 
place, to embrace a paradox, namely, the existence of multiple, 
real times in the universe? Bergson's answer to this question 
inevitably brings us back to his basic philosophic theme, which 
consists of his distinction between real, lived time and its 
"spatialization" into the objects, events, and clock-time of 
everyday life and of scientific activity. According to Bergson, 
our conceptual thinking, as well as its linguistic expression, 
is "molded" upon a world "already made." But our intellect, 
in thus reflecting the world, only serves to mask reality itself, 
that is, the world "in the making," in short, real time or 
duration. Now, given the goal and method of science, physi- 
cists, at least as much as the rest of us, live in a world already 
made and not in the making, a world, therefore, in which 
what is most concrete— time and change— is only superficially 
experienced. "Let us become accustomed," Bergson urges, "to 



see all things sub specie durationis: immediately in our gal- 
vanized perception what is taut becomes relaxed, what is dor- 
mant awakens, what is dead comes to life again." Mathematics 
will not then be "given the status of a transcendent reality"; 
and physicists will no longer be interested in erecting Ein- 
stein's theory, just as it stands, into an unconscious meta- 
physics, one, moreover, that tends in the direction of an ideal- 
ism based upon principles having nothing in common with 
those of relativity. 

As early as 1911, the thesis of the existence of multiple, real 
times in Einstein's theory was dramatized in "the clock para- 
dox of the identical twins." In that year, the eminent French 
physicist Langevin stated before the International Congress 
of Philosophy, meeting at Bologna, that a space-traveler will 
be younger upon his return to earth than his stay-at-home 
twin brother, because not only his time but also his bodily 
processes will have been slowed by the vehicle's motion 
through space. It was hearing this notion of "asymmetrical 
aging," enunciated by Langevin, which, in fact, first drew 
Bergson's attention to Einstein's theory. All of Duration and 
Simultaneity can be considered its refutation, although the 
question is directly treated only on pages 73-79, and in the 
first Appendix, "The Journey in the Projectile." This Appen- 
dixes a reply to another French physicist, Becquerel, whose 
position was the same as Langevin's. Bergson's last word on 
the subject was contained in an article written in 1924 and 
published in reply to one by Andre Metz, a disciple of Bec- 
querel, m which the orthodox view was restated. 

After a lapse of thirty years, the controversy over asymmetri- 
cal aging was reopened in 1956, the principal part in it being 
taKen, this time, by the English astrophysicist, philosopher of 
of T'^ d / Qence edu cator, Herbert Dingle. The criticism 
Prnft!« IT ? as y mmetr ^al aging which is advanced by 
wotessor Dingle rests, like that of Bergson, on the assertion 

translator's preface 


that physico-mathematical "proofs" o£ asymmetrical aging are 
vitiated by Einstein's postulate of relativity. Professor Dingle's 
Introduction to the present work is of great importance in 
itself; and it should serve to heighten the impact of Bergson's 
Duration and Simultaneity upon the intellectual world. 

Leon Jacobson 

July 1965 


Translator's Preface v 

Introduction xv 

Selected Bibliography xliii 

Note on the Translation xlvi 

Duration and Simultaneity 

Foreword to the Second Edition 3 

Preface 5 

Chapter One-Half-Relativity 9 

Chapter Two— Complete Relativity 30 
Chapter Three— Concerning the Nature of Time 44 

Chapter Four— Concerning the Plurality of Times 67 

Chapter Five— Light-Figures 114 

Chapter Six— Four-Dimensional Space-Time 127 
Final Note— Time in Special Relativity 

and Space in General Relativity 157 

Appendix I— The Journey in the Projectile 163 

Appendix II— The Reciprocity of Acceleration 173 

Appendix III— "Proper-Time" and "World-Line" 177 




Early in this century, two very prominent, and originally 
independent, lines of thought collided. The area of impact 
included problems concerned with the experiences, or ideas, of 
time, simultaneity, motion. On the one hand, the chief center 
of interest in philosophy, it is not too much to say, was the 
system of Bergson, in which the passage of time, apprehended 
intuitively, was the fundamental element. On the other hand, 
the physical theory of relativity, which after 1919 at any rate 
dominated scientific thought, submerged time in a more com- 
prehensive and essentially static "space-time," from which it 
could be extracted variously and largely arbitrarily by the 
physicist. It was inevitable that one or other of these views 
should give way. 

As a matter of history, it was the Bergsonian movement that 
yielded. Its influence rapidly waned, and it was succeeded in 
philosophy by ideas of the logical positivist type that origi- 
nated in relativity theory. But is this a final judgment? The 
appearance of Professor Jacobson's very clear translation of 
Bergson's Duree et Simultandite affords an opportunity for a 
reconsideration of the conflict in the light of nearly half a 
century of subsequent research. In this necessarily too brief 
Introduction I shall attempt such a reassessment and try to 
indicate its present significance. 

I should like, however, as a preliminary to reject one type of 
solution to the problem, to which Bergson himself, though he 
specifically disowns it (pp. 64-65), seems at times to resort, 
namely, that of postulating a fundamental distinction between 
philosophy and science. Originally they were one, and although, 
in . the sense in which the words are now used, philosophers 
and scientists may consider different problems and approach 
the same problems from different directions, it is not possible 



that there are two equally valid solutions to the same problem. 
If that were so, discussion would be useless. I shall take it for 
granted that, on the points at issue here, Bergson and the rela- 
tivists might both be wrong but cannot both be right. On that 
basis alone is it worth while to continue. 

Let us begin with the problem which, though not the most 
fundamental, presents the conflict most pointedly-the prob- 
lem of what has come to be known as "asymmetrical aging." 
This is here dealt with at length by Bergson, both in the text 
and in Appendix I. Paul journeys at high uniform speed to a 
distant star and returns two years older, according to his clock 
and his physical condition. Peter, however, who remains on 
the earth, is then some two hundred years older than when 
Paul left him, and has long been in his grave (p. 74). That is 
what, according to the great majority of its advocates, Ein- 
stein s theory of relativity requires. To Bergson, however, time 
lived is an absolute thing, no matter whether it is Peter or Paul 
wno lives it. Hence, however they occupy the interval between 
separating and reuniting, they must live the same time and 
etore age by the same amount. Therefore Bergson has 
condmioT argUment ^ le *ds the relativists to their 

laf^ 0 ^! by dCnyin S that ^ " time " which Peter calcu- 
ates that Paul s clock will record is, in fact, time. It is a "phan- 

u7; a r r> t0 anythin S that Paul experiences. In exactly 
Aerebv ^ Cakulate a P hant °m time for Peter and 

The1a y ct P r e v" 1 U " PetCr whose a g in g h ^ been retarded, 
sivelv rw a rCSUltS arC cont "dictory proves conclu- 

We can H i naples ' j^^ng the other. 

*e the difference of interest that ied to 

tial for Bereson tn g t0 u COnvmce one another. It was essen- 
intuition of ^ the absolute chara «er of time, for the 
took hi standT ^ him ° f the essence oi life; hence he 
tivists' al cu t io r J? ^ had to in ter P ret the rela- 
cakulanon, w hl ch he could not fault, as leading to a 



phantom time. The relativists, however, were not concerned 
with life. To them, Peter and Paul could have been merely 
names of clocks, and all that they claimed was that when the 
clock Paul rejoined the clock Peter, it could be observed to 
have recorded a shorter lapse of time. If, incidentally, there 
happened to be human beings standing by the clocks, they 
would of course age in agreement with their clock readings; 
and if philosophy suggested otherwise, then philosophy was 

But Bergson also advanced a perfectly relevant argument 
even from the physical point of view. To this the relativists 
had no answer, and if he had allowed himself to pose as a 
physicist and left his philosophy out of account, he might have 
been able to press the point home. At the basis of the theory of 
relativity lies the postulate of relativity, according to which, 
when two (or more) bodies are in relative motion, either of 
them can be accorded any motion that one pleases, including 
none at all, provided that the other is then given whatever 
motion is necessary to preserve the relative motion. That 
means that no observation is possible that will enable one to 
say that the motion is divided between the bodies in any par- 
ticular way. But if motion retards the process of aging, the 
relative youth of Paul on reunion would indicate that it was 
Paul, and not Peter, who had moved, or at least had moved 
more, and that would violate the postulate of relativity. Hence 
the theory would require that its own basis was invalid. The 
only possible conclusion, therefore, if the theory was not to 
destroy itself, was that Peter and Paul, whether men or clocks, 
must age at the same rate during the journey. 

This consideration seems to me final. This same problem 
has been revived at various times since it was first conceived, 
and in particular, during the last nine years or so, has been the 
subject of vigorous controversy all over the world. With very 
few exceptions, physicists have maintained that the theory of 
relativity requires asymmetrical aging, notwithstanding the 
argument just given. Some years ago, in an attempt to bring 
the discussion to a point, I put that argument into the form of 


a single syllogism, in the hope that those who did not accept 
its conclusion would state from which of its elements they dis- 
sented, and why, and so enable agreement to be reached. Here 
is the syllogism, as presented in Nature: 1 

1 According to the postulate of relativity, if two bodies (for exam- 
ple two identical clocks) separate and reunite, there is no observable 
phenomenon that will show in an absolute sense that one rather 
than the other has moved. 

2. If on reunion one clock were retarded by a quantity depending 
on their relative motion, and the other not, that phenomenon would 
show that the first had moved and not the second. 

3. Hence, if the postulate of relativity is true, the clocks must be 
retarded equally or not at all: in either case, their readings will 
agree on reunion if they agreed at separation. 

Unfortunately, I underestimated the capacity of the contro- 
versialists for evading the issue. The next contribution to 
Nature 2 began, "May I suggest an alternative approach to this 
problem . . ."; and the writer then proceeded to a relatively 
involved discussion; the syllogism was not mentioned. It is 
hard to see why, when the problem has been reduced to the 
simplest possible terms, a new and indirect approach should 
be necessary; but, in fact, that was but one of an apparently 
endless succession of such approaches. I have repeated the 
syllogism several times, in several places, but without eliciting 
a single answer to the question, which of its elements is faulty, 
and without a single acceptance of its conclusion from any not 
previously convinced of it. 

This is a most remarkable situation, which, quite apart from 
the reality or otherwise of asymmetrical aging, calls for serious 
inquiry. I shall revert to this later: here I shall merely try to 
identify its origin, which I believe lies in the history of the 
subject, while its endurance is facilitated by the unawareness, 
among the younger physicists at least, of that history. It is 
necessary, therefore, to recall its salient features. 

i"The 'Clock Paradox' of Relativity," Nature, CLXXIX (1957), 1242. 
• 2 J. H. Fremlin, "Relativity and Space Travel," Nature, CLXXX (1957), 



From the time of Newton up to the end of the nineteenth 
century, mechanics was regarded as the basic science: his laws 
of motion and their associated equations were the foundation 
on which all further constructions, in the metrical sciences at 
least, had necessarily to be erected. But at the end of the nine- 
teenth century a new possibility was revealed. All attempts to 
establish an electromagnetic theory on a mechanical basis had 
failed; on the other hand, the electromagnetic theory of Max- 
well, amplified and extended from static to moving systems by 
Lorentz, had acquired a character that seemed to qualify it as 
at least a rival to mechanics. Instead of a mechanical theory 
of electricity, an electrical theory of matter claimed the atten- 
tion of physicists; and the Maxwell-Lorentz electromagnetic 
equations vied with the mechanical equations of Newton as 
expressions of the basic laws of the universe. 

Unhappily, these sets of equations were incompatible: one 
could therefore not be derived from the other, so at least one 
had to go. For instance, Newton's third law of motion, that 
action and reaction were equal and opposite, was not possible 
in electromagnetic theory. But the outstanding discrepancy was 
with Newton's first law, or the principle of relativity, as it had 
come to be called. That law implied that a state of uniform 
motion was indistinguishable from another such state and 
from a state of rest. The Maxwell-Lorentz theory, however, 
demanded a static ether with respect to which a moving body 
would exhibit different phenomena from a resting one. Thus, 
between two electric charges, both at rest in the ether, only an 
electrostatic force would appear; but if, though still relatively 
at rest, they were moving in the ether, they would constitute 
two electric currents between which an additional force would 
operate. It therefore became necessary to determine by experi- 
ment whether the various states of uniform motion could indeed 
be distinguished from one another and from a state of rest. Of 
the many experiments devised for this purpose we need con- 
sider only the most famous, the Michelson-Morley experiment, 
discussed in this book. 

This experiment is now usually looked upon as an attempt 



to discover the absolute velocity of the earth, but it was in fact 
much more fundamental than that. It was an attempt to deter- 
mine whether the earth, or any other body, had an absolute 
velocity at all— in other words, whether the Newtonian me- 
chanical theory or the Maxwell-Lorentz electromagnetic theory 
was to survive. The experiment decided against the Maxwell- 
Lorentz theory, and this was Michelson's immediate deduction 
from it. In his paper 3 announcing the result of his first per- 
formance of the experiment he wrote: "The interpretation of 
these results is that there is no displacement of the interference 
bands. The result of the hypothesis of a stationary ether is thus 
shown to be incorrect, and the necessary conclusion follows 
that the hypothesis is erroneous." 

This seemed conclusive, but it had the embarrassing conse- 
quence of depriving electromagnetism of a most successful the- 
ory and leaving nothing in its place. Naturally, therefore, 
strenuous efforts were made to avoid Michelson's conclusion. 
The first comprehensive hypothesis to this end was that of 
Lorentz, who made the ad hoc supposition that motion through 
the ether shortened a body in the direction of motion by a 
certain factor and reduced the frequency of any vibration it 
might possess by the same factor. He showed that if this were 
so, no experiment carried out on any body at all, without 
reference to anything external, could reveal whether that body 
was moving or not (although, in fact, there was a real differ- 
ence between these states) provided that the motion was uni- 
form and that its velocity did not exceed that of light. In 
mathematical terms, the relation between space and time meas- 
urements in relatively moving systems (which became known 
as the "Lorentz transformation") was such that the electro- 
magnetic equations were invariant to it. The relativity ex- 
pressed by Newton's first law of motion was therefore, on this 
view, not a characteristic of nature but a consequence of these 
ethereal effects on moving bodies which operated so as to hide 
from view the real state of motion of a body. 

Shortly afterward Einstein put forward a different theory. 

« American Journal of Science, XXII (1881), 128. 



He was as anxious as Lorentz to save the electromagnetic 
equations, but he was not willing to sacrifice the principle of 
relativity as Lorentz had done. He therefore devised his theory 
of relativity, of which the two basic postulates were the prin- 
ciple (or postulate) of relativity— that all states of uniform 
motion were intrinsically indistinguishable— and the postulate 
of constant light velocity— that light emitted in any direction 
at the same point and at the same instant from each of a num- 
ber of relatively moving bodies moved through space as a 
single beam with a fixed velocity c, the motions of the sources 
having no influence on that of the light emitted. 

This seemed merely to express the original contradiction 
without resolving it. The first postulate granted the validity 
of the mechanical equations, and the second that of the electro- 
magnetic equations, and these were incompatible. But Einstein 
sought a reconciliation by accepting the electromagnetic equa- 
tions, with all their metrical consequences, without accepting 
the ether (that is, anything that could serve as a universal 
standard against which velocity could be measured) which was 
essential to the electromagnetic theory. The rejection of the 
ether made the relativity postulate a reality instead of the 
mere appearance that Lorentz' device had made it, but it laid 
on Einstein the obligation to show how two bodies in relative 
motion could both be moving with the same velocity c with 
respect to the same beam of light. 

He achieved this through the realization of what no one had 
noticed before, that no natural method existed for determining 
the time, according to a given clock, of an event at a distance 
from that clock. Furthermore, he showed that no unambiguous 
determination was possible if his postulates were granted, and 
therefore that the time of such an event had to be defined if it 
was regarded as having any significance. He therefore sought 
a definition that would justify his postulates. Suppose there 
are two points, A and B; and pulses of light, traveling as a 
single pulse, are emitted from sources P and Q when they are 
both at A, P being stationary there and Q moving toward B. 
The light will reach B at some particular instant, at which it 



will be further from P than from Q. An observer with P will 
therefore consider that the light has traveled further than an 
observer with Q, and will therefore accord it a higher velocity 
than the second observer unless the observers allot different 
times to the arrival of the pulse at B. What Einstein succeeded 
in doing was to define a procedure for timing that event so 
that the observers, on applying it, did in fact time the event 
differently and in such a way that they both arrived at the 
same velocity for the light. Moreover, that procedure gave the 
Lorentz transformation for the relation between the times and 
places of events according to observers in relative motion, so 
that, quite independently of Lorentz, he reached just that 
transformation that was needed to preserve the invariance of 
the electromagnetic equations and so to ensure that, if his the- 
ory were correct, no electromagnetic experiment could distin- 
guish between the various states of uniform motion. What 
Lorentz achieved by arbitrarily postulating physical effects of 
the ether on moving bodies, Einstein achieved by arbitrarily 
postulating a certain method of timing distant events. He 
could therefore dispense with the ether and so retain the postu- 
late of relativity as a fundamental fact of nature instead of a 
Slctl° m C ° nSequence of ^ co-operation of different physical 

Since so much has been written in this controversy which 
snows that the writers have not understood Einstein's theory 
t an some of them even think that he discovered the one and 
y natural way of timing distant events instead of inventing 
hSw needCd t0 me the eIe ctromagnetic equations-I quote 
« Jus own summary of the theory,- specially written to cor- 
rect this error, for his lectures at Princeton in 1921: 
The th 

catL,r« n 0 l?l a v , t ^ it ^ i \ often criticized for giving, without justiE- 
« ^^££7? ^ l ° thC PWion of light, in that 
The situation hf P lme u P on the law of propagation of light- 
snuanon, however, is somewhat as follow. In order to give 
'Albert Einxteir. tl 
Nnceton: wS.^. •/ Relativity, trans. E. P. Adam* 

nnceton Umversity Press, 1955), p. 28. 


physical significance to the concept of time, processes of some kind ■ 
are required which enable relations to be established between differ- 
ent places. It is immaterial what kind of processes one chooses for 
such a definition of time. It is advantageous, however, for the theory, 
to choose only those processes concerning which we know something 
certain. This holds for the propagation of light in vacuo in a higher 
degree than for any other process which could be considered, thanks 
to the investigations of Maxwell and H. A. Lorentz. 

This shows beyond question that it is basic to the theory 
that the time of a distant event can be chosen as we wish, and 
that Einstein made his choice in order to justify the Maxweli- 
Lorentz theory. That means, of course, that the only possible 
test of the theory must be kinematical; electromagnetic tests 
will necessarily confirm it since it was framed in order to pass 
them. It must stand or fall (so far as experiment is concerned) 
by the comparison of relatively moving clocks and measuring 
rods to see whether their readings do, in fact, obey the Lorentz 
transformation. No such test has yet been possible, so the the- 
ory remains, like Lorentz', a purely ad hoc device to escape 
from the old predicament. We shall see the significance of this 

Let us, however, return to the historical development. For 
years after these two theories (Lorentz' in 1904 and Einstein's 
in 1905) appeared, they were generally regarded as different 
forms of the same theory since their mathematical content was 
the same, notwithstanding that they were physically fundamen- 
tally different. Einstein's was truly a relativity theory; Lorentz' 
was not, though it had some of the consequences of relativity, 
for example, the impossibility of discovering the state of mo- 
tion of a body from experiments confined to it. Einstein's 
theory extended that impossibility to all experiments. But the 
confusion was accentuated by the fact that, although the the- 
ory was generally accredited to Lorentz (Einstein's name ap- 
pears in connection with it very little before World War I) it 
was given the name "relativity theory," which, in Lorentz' 
form, it certainly was not. Poincare\ for instance, right up to 
his death in 1912, habitually referred to "the relativity theory 


of Lorentz" 5 and scarcely ever, if at all, mentioned Einstein 
in that connection. 

Thus was laid the foundation of a misunderstanding that 
has bedeviled the subject ever since. When Einstein's general 
relativity theory received confirmation at the eclipse of 1919, it 
was universally acclaimed as a logical development of his 
special theory of 1905, and the "relativity" theory then began 
to be ascribed to him alone. But the ideas associated with that 
name (that is, Lorentz' ideas) through the preceding years then 
also went over, with the name, to Einstein. The result was a 
complete confusion. Many physicists regarded the whole thing 
as metaphysical and, despairing of understanding it, contented 
themselves with manipulating the equations, which at any rate 
they could do correctly whatever their meaning might be. The 
"contraction" of moving bodies, for example, which to Lorentz 
(and FitzGerald) was an ordinary physical effect like the con- 
traction through cooling, and to Einstein was merely the result 
of the difference in the times that were regarded as simultane- 
ous by relatively moving observers, was regarded as a single 
conception, but whether it was "real" or "apparent," or 
wnetner there was any longer a difference between reality and 

knSTT'u n ° b0dy exce P l Lorentz an <* Einstein seemed to 
wow. By this time nobody in the mathematical-physical world 
«T . C . are *. <l ue "ion on such a point is immediately taken 
m2Z t 1CatI ° n that the <l uest i°ner does not understand the 
' hC is thereu P° n instructed, politely or sar- 
S a r dmg t0 the d i^sition of the instructor. 

and no7in teT?, etWeen ^ tW ° the ° des is P erfeCtl ? 

the simoW t ^ m y stical - It may be illustrated in 

oncetoasoSn^f ^ ^ ^ 

of relativitv to ° Ck P arad ox," that is, the relation 

docks, A and B T™ 1 ^ 3 ' aging " Su PP° se there arC W ° 
and suppose th*' tlVel ? at rest . at widely separated points, 
to Einstein's andl^ Synchronized with one another according 
lorentz' prescription. For simplicity, suppose 

marion, im)^^'^*™* Poincar ^. DernUres Pemies (Paris: Flam- 



that, if there is an ether, they are at rest in it. Now let a third 
clock, C, be set to agree with A and then moved from the point 
of A to the point of B at high uniform speed. On both theories 
it will read an earlier time than B on arrival. On Lorentz' the- 
ory this will be because its motion through the ether has re- 
tarded its rate of working; on Einstein's theory it will be be- 
cause the definition by which B is set gives it a later time than 
that of C. 

We can now see at once that Lorentz' theory requires asym- 
metrical aging and Einstein's does not. According to the former, 
the working of Paul's clock is actually slowed down by its 
motion through the ether, both outward and back, so that it 
(and Paul) record a shorter time for the journey than Peter 
and Peter's clock which have not moved. On Einstein's theory, 
however, there is no ether to do anything to either clock, so 
each works as though (as in fact is the case on this theory) 
motion made no difference to it. But what the clock at B re- 
cords can have no effect at all on either Peter or Paul. Hence 
there is nothing whatever to require asymmetrical aging, and 
the contrary belief is almost inexplicable. 

I say "almost," and not "quite," inexplicable because it is a 
fact and therefore an explanation must be presumed possible, 
and also because Einstein, who certainly understood his own 
theory, held that belief. The attempt to understand that will 
take us very deep into the heart of the theory itself and show 
us that, notwithstanding its extreme ingenuity and its appar- 
ent success over many decades, it is nevertheless untenable and, 
moreover, could have been seen to be so at the very beginning. 
Its disproof does not rest with experiment or with its mathe- 
matics, but with an inconsistency in the physical part of the 
theory; it has physical implications that are both inescapable 
and incompatible with one another. 

Why, then, did Einstein not realize that his theory pro- 
hibited asymmetrical aging? In the first place, there is evidence 
that, although he recognized its fundamental difference from 
Lorentz', he still thought the observable implications of the 
two theories were identical. In his first paper on the subject he 
thought he had proved that his theory required a moving 


clock not merely to appear to work more slowly than a station- 
ary one but actually to do so; 6 and, moreover, he must have 
seen clearly that unless his theory required everything observ- 
able to be exactly the same as though measuring rods and 
clocks were physically affected by motion, it would be ineffec- 
tive in reconciling mechanics and electromagnetism. Further- 
more, when the Peter and Paul problem was first posed— by 
Langevin in 1911— there were circumstances that prevented it 
from appearing as a serious threat to the relativity theory. To 
begin with, the possibility that velocities sufficient to cause an 
appreciable difference in rate of aging would ever be attained 
was so remote that the problem could not be regarded as other 
than a jeu d'esprit, in quite a different light from that in 
which we see it now. Hence it called for no more than a token 
answer, and this was at hand in the circumstance that, in order 
to return, Paul would have to undergo an acceleration: the 
theory, as it then stood, was applicable only to uniform mo- 
tions and so was not menaced by this fanciful case. 

It is easy to say now that the magnitude of the effect was 
immaterial and that it should have been realized that logically 
an infinitesimal difference in rate of aging was just as fatal to 
the relativity postulate as a very large one. It did not so appear 
then, as I, who remember that time, can testify. We are all 
human beings whose logic is tempered by imagination, and if 
anyone finds it difficult to believe that physicists of genius 
could have put aside a logical point merely because its practi- 
cal implications were negligible, let him reflect on a similar 
case. He probably accepts the statistical interpretation of the 
laws of thermodynamics, which requires that if a kettle of 
water is placed on the fire a large, but finite, number of times, 
me water will sometimes freeze. He accepts this because it does 
not happen. But the theory makes it just as likely to happen 
now as at any other time; suppose, then, he witnesses it to- 
orrow. Will he accept it as just a natural exemplification of 
C statlstl <*l law, or will he look for another cause? At least 

Rekdvifv"^ th3t tW8 Wa$ err °"«>us. »ce H. Dingle, "Special Theory of 
Relativity, Nature, CXCV (1962), 985. 



one eminent physicist, Sir Arthur Eddington, confessed that in 
such a case he would reject the law, which nevertheless he then 
accepted unreservedly. 7 1 do not think we can hold Einstein to 
have been more disingenuous than any other mathematical 
physicist, today or at any time. 

I think, furthermore, that he was acute enough to realize that 
unless Peter and Paul aged asymmetrically his theory failed, 
for the following reason. If they recorded the same time for 
the journey, that time would certainly have to be either two 
hundred or two years. The former would be the requirement 
of Newtonian mechanics, and so could not be that of his own. 
On the other hand, a journey of two years would lead at once 
to an impossibility. It is easy to calculate that Paul's speed 
relative to Peter must be 0.99995 of the speed of light, and the 
distance traveled must be such that light would have taken 
199.99 years to cover it. Hence a beam of light, starting at the 
same time as Paul, would have moved faster all the way and 
yet have returned 197.99 years later-a manifest absurdity. 

It should cause no surprise, then, that Einstein felt that the 
technical removal of this problem from the scope of his special 
theory rendered the problem innocuous. But this escape, of 
course, was no longer possible when later he generalized the 
relativity postulate to cover all motion. 8 He then realized the 
dilemma that faced him: if asymmetrical aging was not a possi- 

7 Sir Arthur Stanley Eddington, New Pathways in Science (Cambridge: 
Cambridge University Press, 1935), chap 3. 

8 Nevertheless during a recent controversy many physicists (for example, 
J. Bronowski, in The New Scientist, Aug. 31, 1961) have continued to 
maintain that Paul's acceleration on reversal prevents the application of 
the special theory to the problem. Curiously enough, however, they do not 
therefore refrain from applying it but regard themselves as entitled to use 
its equations with a meaning of their own in place of that which the 
relativity postulate gives them. The result-need it be said?-is that asym- 
metrical aging is "proved" to follow from Einstein's special theory. The 
reader must be left to appraise this procedure for himself. These writers 
give no sign that they know of Einstein's rejection of such "proof"-or 
indeed of much else in the history of the subject. 


bility, the special theory failed; if it was, the general postulate 
failed. He met this situation 9 by accepting asymmetrical aging 
and invoking "gravitational fields" (using the term in a more 
general sense than the customary one) to save the relativity 
postulate, in the following manner. 

What has to be shown is that Paul will return younger than 
Peter, no matter whether the motion is ascribed to one or the 
other. If it is supposed that Paul moves, he ages slowly, in the 
manner familiar from the special theory, and whatever effect 
the acceleration on reversal might have can be ignored by 
making the journey at uniform motion long enough to pro- 
duce an overwhelmingly greater effect. He therefore ages by 
two years while Peter ages by two hundred. But now, the 
physical conditions being exactly the same, suppose the motion 
is ascribed to Peter, while Paul remains at rest. Then gravita- 
tional fields must be postulated to start, reverse, and stop 
reter, while the operation of Paul's engine-which, in the 
tormer way of speaking, caused his accelerations —now serves to 
Keep rum at rest by neutralizing the effect of the fields. We 
must now consider the influence of the fields on the aging 
process. At the beginning and end, when Peter and Paul are 

iterTt ' thiS influenCC Wil1 be the sa me for both, but on 
evem 0 f motion they are far apart, in regions of different 
^av tanonal potential; and this will make Peter during the 
nZteL T S ° l 3151 ^ C ° m P ared with Pa "l * ™ore than to 

ence fllT' ? Cave him at the ^ with the same differ- 

- ^zizs: d n ?: !h ormer case - The reiativity postuiate 

fields are pure fcdo J v " ltlCWm that SUch g ravitational 
nificant diffe Lee ? ^ M al " C - ThC Sig ' 

but between wT^ T ^ " real " and "fictitious" fields 
at -^^^T fe j like ^ -dings of the clocks 
observable (like th 10n > and wha t is postulated but not 

ingsof se P aLed docSn ati T al fiddS ^ ^ rdatiVe ^ 
«ocks) m order to give a rational description 

9 Albert Einstein "Di 1 
Na turwissenschaften VWiom" Einwii n<le gegen die Relativitatstheorie," 
* ''"'o), 697. 



of the process. It is only the former that the relativity postu- 
late requires to be independent of the standard of rest. 

This argument is, I think, in principle sound and is legiti- 
mately applicable to such a case as that of Foucault's pendu- 
lum, in which the gravitational field of the revolving stellar 
system is called upon to explain the phenomenon when the 
earth is supposed at rest. But it fails here because the observ- 
able phenomena are not the same in the two cases. Suppose a 
clock synchronized with Peter's is placed on the star. When 
Paul is held to move, his clock is behind this one, by approxi- 
mately the same amount, when he reaches and when he leaves 
the star. When Peter and the star are held to move, however, 
the clock on the star is behind Paul's when it reaches him and 
ahead when it leaves him. This is an observable difference, so 
the relativity postulate, which survives a comparison of Peter's 
and Paul's clocks on their reunion, is by this comparison dis- 

This paper of Einstein's seems to be little known: most of 
those who try to reconcile asymmetrical aging with relativity 
use methods that it rules out. The very few writers who adopt 
Einstein's procedure seem to me to have misunderstood it; they 
amplify it in a way which Einstein refrained from attempting 
and which I believe he would have regarded as invalid. A full 
analysis of this treatment of the problem would, I think, afford 
great insight into the nature of the relativity of motion, and 
I have made three unsuccessful attempts to get such an analy- 
sis published. The first two were rejected without assigned 
reason; the third because, it was said, I had "published it all 
before." It would seem that attempts to elucidate this matter 
are held to be necessarily evil, and that their suppression is 
not to be impeded by a misguided regard for accuracy of 

Let us, nevertheless, assume here that my syllogism is sound 
and that in consequence we cannot have both asymmetrical 
aging and the relativity postulate. Then it follows that the 
special theory of relativity must be rejected: if there is asym- 
metrical aging, the relativity postulate, which is essential to 
the theory, is faulted, and if there is no asymmetrical aging, 



then either Newtonian mechanics is valid or Paul covers a 
given distance in a shorter time than a faster beam of light. 
This leads us to seek for the basic error in the theory, for the 
Peter and Paul problem merely shows that there is such an 
error but does not locate it. 

I think the root of the matter can be best seen in terms of 
the Minkowski expression of the theory. According to this, the 
world of nature is represented by a four-dimensional homo- 
geneous mathematical continuum ("space-time"). Everything 
that happens in nature can be analyzed into "point-events"— 
that is, events occurring at single points at single instants-and 
these are represented by points in the continuum. Each such 
point is uniquely definable by four independent co-ordinates, 
which can be chosen in various ways. Each choice corresponds 
to the place (three co-ordinates) and time (one co-ordinate) of 
a pomt-event when a particular standard of position, zero of 
time, and standard of rest are chosen, and any one choice is as 
vahd as any other. The absolute position of the event in space- 
time corresponds to a function of all four co-ordinates which 
is the same for all coordinate systems, and any two events have 
an absolute separation in space-time though their separations 
m space and in time vary with the coordinate system. 

The primary requirement of this theory is that all events 
are analyzable into occurrences at point-instants, and this is 
incompatible with the postulate of relativity. To see this we 
need only consider the simplest possible case, that of two 
ItZlV T* m ° ti0n - Thdr motion is an event, and if it 
eaiTnH °h n Ween ^ and not a P~perty possessed by 
St n f 1V f U Y ' U mUSt ° CCU Py the -hole space needed for 
its manifestation, and that is more than a point The Min- 
kowski concept therefore faik t n , ff a ^ 
of nature ™a . ore tails to afford a true representation 

U^ih™ n n °. maUer WhCther we acce Pt Einstein's or 

in^vi^to' dn ' S thCOry k £ails >> ecause il does 

« Lorent, theory it 

two bodies is made L of tl IVC mOU ° n ' for exam P le * of 



tems are not equivalent. One is unique— that corresponding to 
rest in the ether— and a grained, not homogeneous, space-time 
would be needed to allow for that. 

When this is once realized it becomes a simple matter to find 
cases in which Einstein's theory breaks down. Lest it should 
seem too abstract, however, let us apply it to a particular case. 
Suppose a source of monochromatic light, S, and an observer, 
O, are relatively at rest at a finite distance apart, and let them 
both be provided with synchronized clocks, and O with a spec- 
troscope in which he observes a spectrum line from the light 
of S in a certain position. Now suppose that O moves towards 
S. There are experimental grounds for believing that he will 
at once observe a shift of the spectrum line (the "Doppler ef- 
fect"). But suppose that, instead of O moving toward S, S 
moves similarly toward O at the same clock reading: will O 
observe a spectrum shift at once or later? If the former, an 
effect of an event at S will be transmitted instantaneously to 
O, and if the latter we shall have an observational distinc- 
tion between the motion of O with respect to S and that of S 
with respect to O. Both conclusions are contrary to the special 
relativity theory, yet one of them must be realized. 

The anomaly appears even more strongly if we suppose 
that both O and S move similarly, at the same clock reading, 
in the same direction. If O observes a spectrum shift he can 
calculate a velocity from it, and that must be an absolute 
velocity since there is never any relative motion between the 
only bodies in the system. If he does not observe a spectrum 
shift, the effect of the motion of S must have been transmitted 
instantaneously to him, to neutralize the shift he would un- 
doubtedly have seen if S had not moved. 

This is entirely equivalent to the example that Einstein 
himself took at the beginning of his original paper on the 
subject 10 to show what he meant by his postulate of relativity, 
namely, that in all cases of relative motion the phenomena 
observed are the same whichever body is moved, although the 

10 "Electrodynamik bewegter Korper," Annalen der Physik, XVII (1905), 




ascriptions of the nh» 
moti °n of a magnet anH° mena differ - He took the relative 
ab Ie respects, the curr 3 ° f Wire ' in which ' in a11 observ " 
whichever is moved B, pr ° duced in the wire is the same 
de veloped, that is not « \ * C f° rdin S to the theory which he then 
moved the current is oh b ° dies are far a P art - If the coiIis 
J» moved it i s observed] lmmed ''ately, and if the magnet 
bodies would therefor*^ • • Svnchr onized clocks with the 

11 is e asy to see that l*™^ the 
Phenomenon that if thp , " becaus e we are dealing with a 
demands a finite space T' 7 P ostulat e is true, necessarily 

° r to move, and Z ^ bv its elf can be said to rest 

15 only when at least two hT" Can ban either supposition. It 
mea «ing, and then a Z U * W C ° ncerned that motion has 
£ n X Phenomenon decent rCgl ° n ° f S P ace must be available. 
Doppler effect or the £** ,n & on motion (for example, the 

" b, ^o independent poLT 1 is before irredu- 

M mus t inevitably I T' ^ the attem Pt so to reduce 

in w? V t an ° ther > equal? 1™T - the relat "ity postulate. 
* which the attempt to h^ T' CXam P le >°™ years ago" 

t^T^z^^rr* criterion for syn - 

confrt J ° r COI »pari n? ~i i ♦ , and to the Lorentz trans- 

sstrr that a l at sr ing docks ied to the 

a «d after a than *e othe r N n * WOrk b0th 

ment L I Umber <* re Petif • T ° n ° tlce Was taken of this, 

^^Tb" ,he - -.a — * - 

wron ff nt vT. Born ' 12 who « a , , . cons 'denng came from 

P-ed g an7sh o em - He ^Z L^ 1 ^ adva «- d the 
a n accent WCd tha t it did J ? ° nC 1 should have pro- 
to my P P rohT ° f this a nd a C h embarrass *e theory. Despite 

tiV "y tb eory t T ° btrud es kself 

7 15 SO Nearly untenab e h Wh /' tf the s P ecial ^a- 
u " Kelativ . t S th,s not been realized 

^^It 1 ^' an v ■ 
12 M.Bor„,.. Spec .; e 'XXv„ (1960) 233 a " Ep^stemological Appraisal," 

COr y° f Relate 

y- ^ a / urej cxcvn (lQ63) i2g7 


before? The answer, I think (I shall consider the implications 
of this presently), lies fundamentally in the fact that it has 
become so customary in science to appeal only to experiment 
and not to trust reason, that even the clearest demonstration 
of inconsistency in a theory is ineffectual so long as the theory 
is believed to accord with experiment. The special relativity 
theory has satisfied this condition in numerous instances dur- 
ing the last fifty years, and accordingly it has acquired an 
immunity from rational criticism of which physicists seem 
unaware. In fact, however, there is no experimental evidence 
at all for the theory; all that appears to support it does so 
through a circular argument. To see this the earliest example 
will suffice— the Michelson-Morley experiment. 

In this experiment, as it is invariably described, the times 
taken by beams of light to traverse different paths are com- 
pared, and an explanation is given in terms of the modification 
of these times by the motion of the apparatus. Bergson himself 
(p. 70) accepts this description without question and discusses 
the effect of motion on clocks attached to the apparatus. 

But in fact no clocks at all are used. The experiment is con- 
ducted without reference to a clock or to time, so the effect, if 
any, of motion on clocks cannot account for the observations. 
We observe only interference fringes, which keep a constant 
position throughout. How, then, is time introduced into the 
description? Simply by interpreting the fringes in terms of the 
Maxwell-Lorentz theory which supposes that they are caused 
by light having a constant velocity c, a frequency n, and a wave 
length A, which are related by the equation, c = n\. c and n 
involve time, and so time enters the description. 

But the moment we recall the purpose of the experiment, 
we see that this is quite illegitimate. It was designed to decide 
between Newtonian mechanics and the Maxwell-Lorentz elec- 
tromagnetic theory; we must therefore not presuppose that 
either of these is true. But that is exactly what has been done. 
When the Maxwell-Lorentz theory is presupposed, only two 
explanations are possible: either Newtonian mechanics is 
wrong or there has been some disturbing factor that has been 



s"o?d° k Tt E ^S ChOSC r ** ahe ™ ative and Lorentz the 

autoniatical ySLTafer 1 CXPlanati ° n that MkhdSOn 
is ruled out by the ^ ^T^ 8 ™" ^ * ^ 
it was therefore igno^L^ ex P erime "t is described; 

almost immediatelv Zf byever y° ne e *cept Ritz, who died 
AH the r P po ed . C ° Uld therCfore be for g°«en. 

arguments posino- ac „ aract er; it is a vast assembly of circular 
which it shoConlv a n XPen r mal Verifi -tion of a theory of 
one other example Slder COnsistenc y- T ° take but 

test Einstein's second nnT , ex P eriment s 13 made to 

Maxwell-Lorentz the™ IT ate ~ that > in accordance with the 
that of its source Beam f * ° f Ught h inde P e "dent of 

lent to li ght in this postulate" T" radiation -which is equiva- 
vacuum tube, some held t ^ r ° m hypothetical particles in a 
high speed, i ssue from t ° De stationary an d others moving at 
P ar ed by a highly theoreV , ^ thdr velodt ies are com- 
mferred as equal and J™** 1 technique: the velocities are 
seek for the evidence that * U > ove d." But when we 
ties, we find it in the theJ € ?° Urces have the accepted veloci- 
Proved. If th e velocity of jLV t ***** impIies the thin S » be 
source, that theory is wron " T inde P en dent of that of its 
are meaningless. Further™? ' su PP osed source velocities 

» implied in the descrindnT' ?l qUantum theory (which also 
*e "particles" have nof Z M ** Anient) requires that 
them as sources for this dUaHty necessar y to qualify 

«1 statistics. The whole aT' ^ d ° n0t ° bey d ^ 

f ° m . the only po int 0 f : ie 3 J g t U h m ; nt , is completely confused. 
exper lment is Used ™ that is legitimate when such an 

one source of all the v a Z/l ^ P ° StuIate ' ^ere is but 
t"he Whatever goes S ^uted-the stationary vacuum 
Really, and to * n ^ide can be determined J 

" » obviously fallacious y u Under test to determine 
"For example, that ~* „, 



What, now, are we to conclude from all this concerning 
Bergson's attitude to the relativity theory? In the first place, 
we must recognize that he saw clearly what to nearly all the 
physicists was a matter of confusion, namely, that Lorentz' and 
Einstein's theories were fundamentally distinct. Lorentz' the- 
ory was what he called "half-relativity" or "unilateral rela- 
tivity": Einstein's was "complete relativity" (see especially pp. 
91-92). This, in view of the intellectual climate of the time, 
showed a very clear perception. He had no doubt that, while 
on Lorentz' theory asymmetrical aging was possible— indeed, 
inevitable— it was impossible on Einstein's theory, and it was 
with the latter only that he was here concerned. That in itself 
must have been sufficient to give him confidence that he under- 
stood the matter better than the physicists, to whom the equa- 
tions were the essential thing and their meaning relatively 

On the other hand— whether through modesty or oversight 
of what must have appeared to him unimportant compared 
with his intense awareness of the vital character of time and 
the inertness of space— Bergson was willing to grant the physi- 
cists everything they claimed that did not directly menace his 
own philosophy. Insofar as time had spatial qualities he was 
willing for it to be spatialized, and so he failed to see the 
inherent contradictions in the special relativity theory that 
would have made it unnecessary for him to defend his phi- 
losophy against it. In that defense he accordingly used reason- 
ing that failed to convince the physicists because it missed the 
point to which they attached importance. Since the time that 
Peter ascribed to Paul was not the time that Paul lived, he 
called it a phantom, that is, something unreal, because, as he 
insisted, only what is perceptible is real (for example, see p. 
108), and he likened it to the diminished size which a distant 
object seems to possess but which corresponds to nothing ob- 
servable at the position of that object. The analogy is good 
up to a point, but it breaks down precisely where it is most 
needed. The physicist could retort-and in these days of auto- 
mation the retort comes even more readily to mind— that all 


dtr Id m r m ° Ved fr ° m ^ P^dings. Properly adjusted 
docks could be stationed at the points considered; the iavel- 
ng clock, having been set, could be moved mechanically; and 

examm £ ?" * " - 
Paul', rU t ™ S by an y° ne at leisure: the readings of 

* cici 5. 

between 2™ for Ber S son to claim a distinction 

that even ll? P hllos °Phy did not necessarily demand 

ldl> even between two event-: ,t,i,<, t i • 

persons present at Jwi? , C Same P lace ' two hving 
would douhS I mUSt have " lived " the ^me time. He 
^t^?-"^"? that ' if one of them had taken 
«1 state miSt b ' mental ex P erien ces and his physiologi- 

*an that ^rT^ ^ 3 ^ ° f ^ 

effect, why might ' ° her ' and if drugs could produce that 

would, and he w " L ° rentz ' theor y k certainly 

time," in Bergson' 38 C ° ntestin & Lorentz' theory. "Living 
vi dual, and a con/ SenSC ' IS necessaril y an attribute of an indi- 
individuals can s h am ° n between the living times of different 
concerned with it. °™ nothin & tha t invalidates a philosophy 

We ^e the samV" S1C n3tUre al °" e - 
of the two simult Unnecessar y deputation in the discussion 
Bergson and Einstein 6 "' 68 "' intuitive " and "learned." Here 
The only absolute si " 7^' faCt ' in com P Iete agreement. 
ev ents at the same iT y ' accor ding to both, was that of 
Einstein's theory-an/^' ^ W3S 3 basic requirement of 
sight-that the simult strikin g evidence of his in- 

3 matter of voluntarv ri"fi ° f se P arated ev ents could be only 
here between them a J 0 "' There was no difference at all 
of view, Bergson stres^S' u ° Wing t0 their difIerent P ointS 
dent events and Einste I. intuitive simultaneity of coinci- 
dents. They were simn! simuI taneity of separated 
*J same coin. y caIIln g attention to different sides of 

an evaluatio7of U BeS!' °> WUhin my cor npetence, to attempt 
knowledge, but I think ? * philoso P h y in the light of present 
" may be said that, beyond doubt, it is 


no longer menaced by the physical considerations against 
which he defends it in this book and which may well have 
been responsible for the fall in esteem that it suffered as the 
relativity theory became established. Indeed, we may go further. 
I think there can now be no doubt that the "space-time," 
which seemed to Bergson on philosophical grounds to be 
merely an artificial construction, is in fact just that. The many 
mystical ideas that have been built on the supposed discovery 
that there is in nature some objective thing called "space- 
time," while space and time are merely the subjective products 
of our arbitrary analysis of this "reality"-these ideas can now 
be dismissed as purely fictional. "Space-time" is a mathematical 
conception formed by combining the co-ordinates (x, y, z, t) 
occurring in the electromagnetic equations. How those co-ordi- 
nates are in fact related to our measurements of space and 
time remains to be discovered, but we can say with certainty 
that they cannot be identified with those measurements. If 
Lorentz' theory is correct, they correspond to the readings of 
distorted instruments, and it is the distortions, and not the 
quantities that we try to measure, that are related with one 
another in the supposedly inseparable way. If, on the other 
hand, the electromagnetic equations are fundamentally wrong, 
then "space-time" is merely a characteristic of a false theory- 
a conception needed to preserve that theory from immediate 
disproof. Only further experiment can tell us which of these 
alternatives is correct, and the most promising of such experi- 
ments would be a properly designed determination of the rela- 
tion of the velocity of light to that of its source. 

We still await the performance of such an experiment, but 
there is no doubt about the attitude of Bergson to this situa- 
tion: he would certainly have expected Lorentz' theory to be 
disproved. Another way of expressing the choice, as we have 
seen, is that it lies between a nonrelativistic world in which 
motion can be analyzed into a succession of points occupied at 
successive instants (Lorentz* theory) and a relativistic world in 
which motion is not so analyzable. Bergson emphatically fa- 



voured the second alternative, 14 and he would therefore have 
been compelled to reject Lorentz' theory. The relativity postu- 
late, on the other hand, while perhaps not essential to his 
philosophy, is in complete harmony with it. When the neces- 
sary experiment is performed, therefore, it should provide 
some real physical evidence concerning the Bergsonian philos- 
ophy in place of the false attack he had to meet. 

Turning from the future to the past, however, we may say 
that in one fundamental respect the influence on philosophy 
of the schools generated by relativity theory has been un- 
fortunate. Bergson was concerned with experience, as essential 
philosophy must ever be-in his case pre-eminently with the 
experience of the passage of time. Physics also is concerned 
with experiences, but with relatively trivial ones, that is, those 
amenable to measurement." But the effect of the relativity 
theory on philosophy has been to concentrate attention on the 
instruments used to represent experiences by concepts-in par- 
ticular, languages-as though they were the ultimate objects of 
philosophical thought. This is the counterpart of the situation 
m science, in which mathematics is in the saddle and rides 
physics, so that, for example, Lorentz' and Einstein's theories 
are thought to be identical because they have the same mathe- 
matical structure. The only difference is that while the lin- 
guistic philosophers allow their symbols to say nothing, the 
mathexnaticians make theirs talk nonsense. This is not to decry 
the study of anguages-it is a necessary study-but when we 
allow it to release us from the duty of saying something until 

Hkel rj ^ ^ FOblemS the ? P res " nt > -hi* in all 
ikelihood we shall never do, we go badly astray. If only as a 

zn::^ a revival of interest in ~ S 

tra^.?^^ T t n f. h Ber ^' An " Metaphysics, 



There is, however, another, still more serious, issue raised 
by the history of the relativity theory, which is of such vital 
concern to us all that it cannot pass unnoticed in this connec- 
tion though I can only touch on it very briefly here. Science 
(and philosophy also, for in this respect they are alike) depends 
on complete obedience to the demands of experience and 
reason. We must accept whatever experience reveals to us, and 
the theories we form to rationalize it must be logically im- 
peccable. In principle this has always been acknowledged, but 
in science, because of its history— modern science began largely 
as a revolt against the undue neglect of experience in philoso- 
phy—the assessment of theories has been left almost entirely to 
experience. Imagination has been allowed to lead the theo- 
retical scientist into various fields of conceivability, notwith- 
standing that no proof is immediately available that they are, 
in fact, realizable in experience. Hence the scientist has not 
been dismayed, but rather exhilarated, by the co-existence of 
mutually incompatible theories concerned with the same set 
of phenomena, because he has had implicit faith in the ability 
of experience (observation or experiment) ultimately to reveal 
which of them is false. 

The method is ideal, so long as the time available is un- 
limited and the experiments harmless. Indeed, so perfect is it 
that there has been no need to examine the internal structure 
of a theory with much care: give it rein, whatever it might be, 
and experience will ultimately dispose of it if it is unsound. 
It is true that some theories can be ruled out at once because 
they are internally inconsistent, for although no theory can be 
proved right by reason alone, it might be proved wrong by 
reason alone. But science— in the past perhaps with much wis- 
dom—has thought it better to let wheat and tares grow to- 
gether until the harvest than to risk destroying wheat through 
a premature purification. Accordingly, there has developed in 
the scientific world an attitude of tolerance toward fanciful 
speculations, especially if they are adorned by an array of 
mathematical formulae, which might in the future acquire a 


support from experience that they cannot yet claim, and at the 
same time an unwillingness to abandon theories that have 
proved useful, no matter what logical defects they might con- 
tain. However reprehensible this might appear from a detached 
philosophical point of view, it has had at least the justification 
of assured ultimate success. 

The momentous fact, however, which is not yet realized, is 
that within the last generation this method has ceased to be 
permissible. The fanciful speculations just referred to, which 
are most evident at the present time in the field of cosmology, 
are of relatively slight importance. They merely waste time 
and money and mislead the public harmlessly for a time on 
matters in which the interest of the public is ephemeral; they 

dal . 7 ^ enj ° y thdr f3nfare and no -reparable 
damage is done. The continued adherence to logically dis- 
provable theories, however, is another matter. Certainly, ex- 
lut™^ " reaS ° n ' bC trUSted t0 dis P— them; 

d^; o a The TT ent . being what the y now are ' th£ y ™y 

on he o C f J , ' T J ° r Catastr °P he - To experiment now 

shall Jel n * ^ that Truth is P*" and 

t 1 1 °r f rl °° k ° nC -^n thishappens 

what ™ J £ , l ° ^ " k Prevails or not - Y « that is 
what now goes on dady m our research establishments. 

it fa manifelTf ^ ^ ° Pini ° n ° r a theoretical Possibility; 
XylZtoZ H ' acknowled S ed by the mathematical 

o "wha Ttht"f y T th Tf' 1 am SUrC ' Wkhout the ^ast idea 
Z^^^^":™^ ~ that the 
prediction, or tl > 1 7 heart of modern scientific 

i^r;:^; z£?£r development in physics 

proof that ft is false h^eefgiven * * ^ A 

answer-from Professor Mav ? u ° nIy auth ontative 
generally acknowtdg ed as thf^ j'V ^ * 

cists-is to this effect: "The sLnTf ******** ^ 

tween soace rnnrH „ , j P C fact that a11 relations be- 
tween space co-ordinates and time exnre*^ u >u r 

transformation can be reoresentPH pressed b ? ^ Lorentz 



contradiction in the theory." 16 In other words, the fact that a 
piece of algebra corresponds to a piece of geometry is sufficient 
to guarantee the tenability of a theory; what the algebraic 
symbols or the geometrical figures mean in terms of experience, 
of observation, is irrelevant. On the same principles one could 
say: the simple fact that the equation y = ax + b can be repre- 
sented geometrically by a straight line should suffice to show 
that there can be no logical contradiction in the Aristotelian 
theory that the path of a projectile is rectilinear. The success 
of range finding conducted on this basis would give a clear 
indication of what we are to expect in the not too distant 
future. There is now no reason at all for doubting that mate- 
rial velocities exceeding that of light are possible and may well 
be attained before long. In terms of the special relativity the- 
ory, however, they will be automatically underestimated. What 
may happen is anybody's guess. 

This situation is a natural, though not an inevitable, devel- 
opment from that which faced Bergson. The danger, which I 
think he saw instinctively but was not able effectively to avert, 
was that of mistaking ideas for experiences, symbols for obser- 
vations. But at that time it was clearly seen by both sides that 
the relation of symbols to experience was an essential part of 
the theory, and if it had then been shown, from physical con- 
siderations, that Paul would not in fact have aged in the man- 
ner that the symbols indicated, the theory would by common 
consent have been abandoned. That is not so today. Physical 
considerations now count for nothing; the mathematics is all. 
If a symbol is given the letter t, then our experiences of time 
must necessarily follow the course that the symbol takes in the 
logically impeccable theory. 

And nobody minds. Not a single dissentient voice has been 
raised in response to Professor Born's ruling, and one must 
conclude— as is indeed evident from other considerations 17 — 

16 M. Born, "Special Theory of Relativity," 1287. 

IT For a few of many examples, see Samuel and Dingle, A Threefold 
Cord (London: Allen & Unwin, Ltd., 1961). 


that it is the general guiding principle of those who hold our 
lives in their hands. I have tried to direct attention to the 
danger inherent in this situation, but without success; my 
attempt to bring it to the attention of the potential victims 
has been refused publication by both the scientific and the 
nonscientific press-the latter understandably, since it must be 
almost impossible for the layman to believe that the scientist, 
whose reputation for absolute integrity has become proverbial, 
can really behave in such a way. Yet it is manifestly so, as any- 
one who cares to read the literature can verify for himself. 

The facts must be faced. To a degree never previously 
attained, the material future of the world is in the hands of 
a small body of men, on whose not merely superficially appar- 
ent but absolute, intuitive (in Bergson's sense of the word) 
integrity the fate of all depends, and that quality is lacking. 
Where there was once intellectual honesty they have now 
merely the idea that they possess it, the most insidious and the 
most dangerous of all usurpers; the substitution is shown by 
the fruits, which are displayed in unmistakable clarity in the 
facts described here. After years of effort I am forced to con- 
clude that attempts within the scientific world to awaken it 
from its dogmatic slumber are vain. I can only hope that some 
reader of these pages, whose sense of reality exceeds that of 
the mathematicians and physicists and who can command suffi- 
cient influence, might be able from the outside to enforce 
attention to the danger before it is too late. 

Herbert Dingle 

April 1965 

Selected Bibliography 

Adolphe, Lvdie. L'univers bergsonien. Paris: La Colombe, 

Becquerel, Jean. "Critique de l'ouvrage Duree et simultani- 
ite," Bulletin scientifique des etudiants de Pans, X (March- 
April 1923), 18-29. 
Bergson, Henri. "Analyse de l'ouvrage de Guyau La Genese 
de I'idee de temps," Revue philosophique de la France et de 
Vetranger, XXXI (1891), 185-190. 

. creative Evolution. Trans. Arthur Mitchell. New 

York: H. Holt Co., 1911. 

. The Creative Mind. Trans. Mabelle L. Andison. New 

York: Philosophical Library, 1946. 

An Introduction to Metaphysics. Trans. T. E. Hulme. 

"The Library of Liberal Arts," No. 10. New York: The 
Liberal Arts Press, Inc., 1955. 

. Laughter. Trans. C. Brereton and F. Rothwell. New 

York: Macmillan Co., 1911. 

. Matter and Memory. Trans. N. M. Paul and W. J. 

Palmer. New York: Macmillan Co., 1911. 

. Mind-Energy. Trans. H. Wildon Carr. New York: H. 

Holt and Co., 1920. 

"Remarques sur la theorie de la relativite" (Minutes 

from the session of April 6, 1922). Bulletin de la SocUti 
francaise de Philosophic XVII (April 1922), 91-113. 

-Second Reply to Second Letter of Andre Metz," Revue 

de'philosophie, XXXI (July-August 1924), 437-440. 

. "Les Temps Fictifs et le Temps Reel" (First letter in 

reply to letter of Andre Metz), Revue de philosophic XXXI 
(1924), 241-260. 

. Time and Free Will. Trans. F. L. Pogson. New York: 

Macmillan Co., 1913. 



• The Two Sources of Religion and Morality. Trans. R. 

Ashley Audra and Cloudsley Brereton. London: Macmil- 
lan and Co., Ltd., 1935. 

Berteval, W. "Bergson et Einstein," Revue philosophique de 
la France et de Vetranger, CXXXII (1942), 17-28. 

Berthelot, Rene. "L'espace et le temps chez les physiciens," 
Revue de Metaphysique et de Morale, XVIII (1910), 744- 

Busch, J. J. "Einstein et Bergson, convergence et divergence 
de leurs idees," Proceedings of the Tenth International Con- 
gress of Philosophy, ed. E. W. Beth and H. J. Pos. Amster- 
dam: North Holland Publishing Co., 1949. 

Caper, Milic. "La theorie bergsonienne de la matiere et la 
physique moderne," Revue philosophique de la France et 
de Vetranger, LXXVII (1953), 30-44. 

Chevalier, Jacques. Henri Bergson. Trans. Lilian A. Clare. 
New York: Macmillan Co., 1928. 

Crawford^ Frank S., Jr. "Experimental Verification of the 

« ^°oL 9 r e ad o° X ' ° f Relativit y-" Mature, CLXXIX (January 
o, iy57), 35—36. 

Dingle, Herbert. "The 'Clock Paradox' of Relativity," Na- 
ture CLXXIX (April 27, 1957), 1242. This article is a reply 
to Crawford (noted above). 

~~n' " The ^ Iock Paradox of Relativity," Science, CXXVII 

MnZ\ ' \ 9 t 8 \' 15fM57 - This anide is a -P^ ^ Mc- 
Millan (noted below). r 7 

• "Relativity and Space Travel" Nature CT XXVTT 
(April 28, 1956), 782-78^. See McCrae (ZZ '^ 

~Th2v7T IT 11 - P ? il0S °P hic ^plications of the Special 
Vd p elat T? m Alb <« Einstein: PhUosopher-Sci- 
bra v o -. PAUL p A .™ R Sc ™- Evanston, 111, The Li- 
brary of Livmg Philosophers, 1949. P p 537-554 

\^mt nce l ° Phll ° sophy - ° xfOTd: The 


Einstein, Albert. Relativity: The Special and General The- 
ory. Trans. Robert W. Lawson. New York: Peter Smith, 

. The Meaning of Relativity. Trans. E. P. Adams. Prince- 
ton: Princeton University Press, 1955. 

Guillaume, Edouard. "La question du temps d'apres Berg- 
son," Revue ginerale des sciences, XXXIII (October 30, 
1922), 573-582. 

Heidsieck, Francois. Henri Bergson et la notion d'espace. 

Paris: Le Circle du Livre, 1957. 
Jankelevitch, Vladimir. Bergson. Paris: F. Alcan, 1931. 
Langevin, Paul. "L'Evolution de l'espace et temps " Revue 

de Metaphysique et de Morale, XIX (1911), 455^66. 
Lovejoy, Arthur O. "The Paradox of the Time-Retarding 

journey," The Philosophical Review, XL (1931), 48-68 and 


McCrae, W. M. "Criticism of Herbert Dingle's Article Rela- 
tivity and Space Travel,'" Nature, CLXXVII (April 28, 
1956), 783-784. 

McMillan, Edwin M. "The 'Clock Paradox' and Space 

Travel," Science, CXXVI (August 30, 1957), 381-384. 
Metz Andre. La relativity expose sans formules des theories 
d'Einstein et refutation des erreurs contenues dans les 
ouvrages les plus notoires (Duree et simultaneite). Preface 
by Becquerel. Paris: E. Chiron, 1923. 

" Le Temps d'Einstein et la philosophic: a propos de 

l'ouvrage de M. Bergson, Duree et simultaneiU," Revue de 
philosophic XXXI (1924), 56-58. 
Voisine, G. "La duree des choses et la relativite. A propos 
d'un livre recent de Bergson," Revue de philosophic XXII 
(1922), 498-522. 
Watanabe, Satosi. "Le concept de temps en physique mo- 
derne et la duree pure de Bergson," Revue de Metaphysique 
et de Morale, LVI (1951), 128-142. 

Mote on the Translation 

The present translation is taken from the fourth edition of 
Duree et Simultaneity as published by the Librairie Felix 

"7,™ f ° Urth editi ° n is a re P rint of the second 
edition of 1923, which must be considered Bergson's definitive 

text All Bergson's footnotes have been translated; footnotes 

in brackets are those of the translator and are intended to 

danfy Bergsons text The mathematical formulae and the 

diagrams are taken directly from the French text 



Foreword to the Second Edition 

The text of this second edition is the same as that of the 
first, but we have added three Appendixes intended to over- 
come certain objections or, rather, to correct certain misunder- 
standings. The first Appendix has reference to "the journey in 
the projectile," the second, to the reciprocity of acceleration, 
and the third, to "proper-time" and "World-line." Despite the 
diversity of their titles, all three are concerned with the same 
subject and reach the same conclusion. They plainly demon- 
strate that, as far as time is concerned, there is no difference 
between a system endowed with any motion whatever and one 
in uniform translation. 



A few words about the origin of this work will enable the 
reader to understand its purpose. We began it solely for our 
own benefit. We wanted to find out to what extent our con- 
cept of duration was compatible with Einstein's views on time. 
Our admiration for this physicist, our conviction that he was 
giving us not only a new physics but also certain new ways of 
thinking, our belief that science and philosophy are unlike 
disciplines but are meant to implement each other, all this 
imbued us with the desire and even impressed us with the duty 
of proceeding to a confrontation. But our inquiry soon ap- 
peared to hold more general interest. Our concept of duration 
was really the translation of a direct and immediate experi- 
ence. Without involving the hypothesis of a universal time as 
a necessary consequence, it harmonized quite naturally with 
this belief. It was therefore very nearly the popular idea with 
which we were going to confront Einstein's theory. And the 
way this theory appears to come into conflict with common 
opinion then rose to the fore: we would have to dwell upon 
the "paradoxes" of the theory of relativity, upon multiple 
times that flow more or less rapidly, upon simultaneities that 
become successions, and successions simultaneities, whenever 
we change our point of view. These theses have a clearly defined 
physical meaning; they state what Einstein, in an intuition of 
genius, read in Lorentz' equations. But what is their philo- 
sophical meaning? To get at this, we went over Lorentz' for- 
mulae term by term, seeking the concrete reality, the perceived 
or perceptible thing, to which each term corresponded. This 
examination gave us a quite unexpected result. Not only did 
Einstein's theses no longer appear to contradict the natural 
belief of men in a single, universal time but they even corrobo- 



rated it, accompanied it with prima facie evidence. They owed 
their paradoxical appearance merely to a misunderstanding. 
A confusion seemed to have arisen, not in the case of Einstein 
himself, to be sure, nor among the physicists who were making 
use physically of his method but among some who were giving 
dus physics, just as it stood, the force of a philosophy. Two 
different conceptions of relativity, one abstract and the other 
full of imagery, one incomplete and the other finished, co- 
existed in their minds and interfered with one another. In 
"2 Up f ^ 1S conf ™°n, we did away with the paradox. It 

dZ LZ J° reP ° rt thiS - We Would thus be helping to 
clear ^ u p the theory of relativity for the philosopher. 

had fel f m0 hr r dse ' the anal y sis which we 

* n A , l ? P roceed mad e the salient features of time 

shar D v i?* m C F hyS ' ldSt ' S calcul ^ions stand out more 
we had" of Tu ° Ut l ° C ° mplete ' not J' ust -nfirm, - hat 
ne*w t „,i k S um * dur ation. No question has been more 
Se n de y rl P hll ° SOph ? S than that of <™e; and yet they all 
STterifST^v 1 - C3pital ira P°" an ^ This'is because 

^^^^VT'^l' having thor " 

treat the other similar^ B a ll' " * " Z " 

that way. The analog/ herein dm! T.T™ " 

whollv extpms.1 o j ""ween time and space is, in fact, 

^t^^,^? 1 * is the ^ of our using 

analogy, S f^^oTf ™ « ^ 
like those of space frT, ? g mg m time £or features 
that covers til a 'nH * pW ^ WC sha11 st °P' at *P ace 

ence-we l a Hot h ^ il visuall y f - our conveni- 
we not g^n L" ve r in PU ^ ^ * time itscIt What ™ uId 
difficult pJ^r^' ™ rr ipt ? ril « it! The ^ ^ the most 

extended^n X fn Z^T ^ ^ We had in the P ast 

has supped us J^^T ^ ^ ° f 

it a bit further. IOr resun "ng it and carrying 

e ' u P° n a cl early delimited subj ect. 



We have carved out of the theory of relativity that which con- 
cerns time; we have laid the other problems aside. We thus 
remain within the framework of special relativity. Moreover, 
the theory of general relativity is itself about to enter there, 
when it wants one of the co-ordinates to represent actual time. 



The Michelson-Morley experiment; half or "unilateral" 
relativity; concrete meaning of terms entering into the 
Lorentz formulae; expansion of time; breakup of simul- 
taneity; longitudinal contraction 

The theory of relativity, even the "special" one, is not exactly 
founded on the Michelson-Morley experiment, since it expresses 
in a general way the necessity of preserving a constant form 
for the laws of electromagnetism when we pass from one sys- 
tem of reference to another. But the Michelson-Morley experi- 
ment has the great advantage of stating the problem in concrete 
terms and also of spreading out the elements of its solution 
before our very eyes. It materializes the difficulty, so to speak. 
From it, the philosopher must set forth; to it he will continu- 
ally have to return, if he wishes to grasp the true meaning of 
time in the theory of relativity. How often has not this mean- 
ing been described and commented upon! Yet it is necessary 
that we do so once more, for we are not going to adopt straight 
off the interpretation given it today by the theory of relativity, 
as is usually done. We want to save all the transitions between 
common-sense time and Einstein's. We must therefore replace 
ourselves in the state of mind in which we were to be found 
in the beginning, when we believed in a motionless ether in 
absolute rest, and yet had to account for the Michelson-Morley 
experiment. We shall thus obtain a certain conception of time 
which is half-relativist, one-sided, not yet Einstein's, but with 
which we consider it essential to be acquainted. The theory of 
relativity may ignore it as much as it likes in its properly 




scientific inferences; it still undergoes its influence, we believe, 
as soon as it stops being a physics to become a philosophy. 
This, it appears to us, is where those paradoxes, which have so 
alarmed some, so beguiled others, come from. They stem from 
an ambiguity. They arise from the fact that two mental views 
of relativity, one radical and conceptual, the other less thor- 
oughgoing and full of imagery, accompany each other in our 
minds without our realizing it, and that the concept undergoes 
contamination by the image. 

Let us then schematically describe the experiment set up by 
the American physicist, Michelson, as early as 1881, repeated 


Figure 1 

by tan and Morley in 1887, and recommenced with even 
greater care by Morley and Miller in 1905. A beam of light 
L n,T } v 0m , S ° UrCe S " divMed ' at Point O, by a thin 
fnS two b m ^ " " angk ° f 45 ° to the beam' direction, 
io in l T 8 ° £ Whkh iS reflected Perpendicularly from 
Lolt nT° B > While the other ^tinues along the 
«,r.™ 1 ° n j °- At Points A and B, which we shall 

oTand 0« T °' ^ tW ° mirrors Perpendicular to 

relctLw J, ^ ^ ' efleCted from mi ™s B and A 

~« ^ough the glass 
reflected from X* i 7 P rolon g atl on of BO; the second is 
uper moosed t " g ^ S3me line OM - The y are thus 
Tb oS I"! g 3 SyStCm ° f in ^erence bands which 
be observed from po in t M in a lens sighted along MO. 



Suppose for a moment that the apparatus is not in translation 
in the ether. It is evident at once that, if the distances OA and 
OB are equal, the time taken by the first beam to travel from 
O to A and return is equal to the time taken by the second 
beam to travel from O to B and return, since the apparatus is 
motionless in a medium in which light is propagated with the 
same speed in all directions. The appearance of the interfer- 
ence bands will therefore remain the same for any rotation of 
the device. It will be the same, in particular, for a 90° rotation 
which will cause OA and OB to change places with one 

But, in reality, the apparatus has been involved in the earth's 
orbital motion. 1 It is easy to see that, this being so, the double 
journey of the first beam ought not to take as long as the 
double journey of the second. 2 

Let us indeed calculate, by the usual kinematics, the dura- 
tion of each of the double passages. With a view to simplifying 
the exposition we shall grant that the direction SA of the beam 
of light has been so chosen as to be the same as that of the 
earth's motion through the ether. We shall call v the speed of 
the earth, c the speed of light, and I the common length of the 
two lines OA and OB. The speed of light with respect to the 
apparatus will be c-v in the passage from O to A. It will be 
c + v for the return. The time taken by light to go from O to A 

I I , . 

and back again will then be equal to + , that is, to 

6 n c-v c +v 


^, and the path traversed by this beam in the ether to 

c* - v z 

2lc 2 21 

; or r. Let us now consider the passage of the beam 


1 The earth's motion may be thought of as a rectilinear, uniform trans- 
lation during the course of the experiment. 

2 It will not do to forget, in all that is about to follow, that the radi- 
ations emitted from source S are immediately deposited in the motionless 
ether and are, consequently, in terms of their propagation, independent 
of the motion of their source. 



that goes from the glass plate O to the mirror B and returns. 
Since the beam of light is moving from O to B at speed c, but, 
on the other hand, the apparatus is traveling at speed v in the 
direction OA perpendic ular to OB, the relative speed of the 
beam of light is now yj^fl; and, consequently, the time 

taken for the entire distance covered is 


. This is what 

y - v 2 

we would see again, without directly considering the composi- 
tion of speeds in the following manner. When the beam re- 
turns to the glass plate, the latter is at O' (figure 2) and the 
beam has touched the mirror when the latter was at B', the 
mangle OB O' being, moreover, plainly isosceles. Let us then 

Figure 2 

stance covered m the 0B >O> passage ^ ^ ^ same 
a * e 00 , d . tance cQvered) ^ ^^^^ ^ 

is 0P c v 

transferri " '• ^ ^ ^ ^ ^^^^ we obtain ' by 
ans errmg mto this last equality the value of OP derived 
from the first: OB' = lc 

^?rrp" Th e time for the distance covered 

- , Zl . and the dis- 

over line Ofi'O' ic t u„ f 

« therefore indeed 



2lc 21 

tance actually covered in the ether, or mis 

\ c 2 

amounts to saying that the earth's motion through the ether 
affects the two passages differently and that if a rotation im- 
parted to the device leads its arms, OA and OB, to change 
places with one another, a shift in the interference bands 
ought to be observed. But nothing of the sort happens. The 
experiment, repeated at different times of the year, for differ- 
ent speeds of the earth with respect to the ether, has always 
given the same result. 8 Things happen as if the two double 
passages were equal, as if the speed of light with respect to the 
earth were constant, in short, as if the earth were motionless 
in the ether. 

Here, then, is the explanation offered by Lorentz, one that 
also occurred to another physicist, Fitzgerald. According to 
them, the line OA would contract as the result of its motion 
in such a way as to re-establish equality between the two 
double passages. If the length of OA, which was J when at rest, 

becomes l^jl - ^when this line moves at speed v, the distance 
covered by the beam through the ether will no longer be meas- 
21 i ^ 2J 

* — . — x 

found equal in actuality. It is therefore necessary to assume 
that any object moving with any speed v undergoes a contrac- 
tion in the direction of its motion such that its new dimension 

is to the old in the ratio of yjl^ 2 to unity. Of course, this 
contraction overtakes the ruler with which we measure the 
object as well as the object itself. It thus escapes the terrestrial 

3 It has been carried out under such precise conditions, moreover, that 
any difference between the two passages of light could not fad to appear. 

ured as -^-y, but as—^— , and the two passages will be 
1-72 \/*-75 



observer. But we would become aware of it if we were in a 
fixed observatory, the ether. 4 

More generally, let us call S a system motionless in the ether, 
and S', another example of this system, a double, which was 
first at one with it and then broke away in a straight line at 
speed v. Immediately on parting, S' contracts in the direction 
of its motion. Everything not perpendicular to its direction of 
motion shares the contraction. If S was a sphere, S' will be an 
ellipsoid. This contraction explains why the Michelson-Morley 
experiment gives the same results as if light had a constant 
speed equal to c in all directions. 

But it is also necessary to know why we ourselves, in our 
turn, measuring the speed of light by terrestrial experiments 
such as those of Fizeau and Foucault, always get the same fig- 
ure c no matter what the earth's speed may be with respect to 
the ether. 5 The observer motionless in the ether will explain 
it thus: in experiments of this type, the beam of light always 
makes the double trip of departure and return between point 
0 and another point, A or B, on earth, as in the Michelson- 
Morley experiment. In the eyes of the observer who shares the 
earth's motion, the distance of this double journey is therefore 
21. Now, we say that he always finds the same speed c for light. 
Always, therefore, the clock consulted by the experimenter at 

Point O shows that the same interval /, equal to 2 J, has elapsed 

between the departure and return of the beam. But the ob- 

t^HT 01 at ,° nCe 0131 instead of a longitudinal contraction, a transverse 
expans, on could just as well have been assumed, or even one or the other 
been nh,' m ^ pr °P° rtion - Regarding this point, as many others, we have 
^een obhged to bypass the explanations given by the theory of relativity. 

5 It 1 ;T g ° UrSelVeS t0 what concerns our present inquiry, 
contrar inn m P ortant l ° note (though often omitted) that the LorenU 
ZlTr " 0t 6nOUgh t0 CStablish . f™m the standpoint of the ether, the 
w7must Michcl »n-Morlcv experiment performed on earth. 

tanehira f of t0 K U , the of time and the breakup of simul- 

theory Thl u " We Sha " re<li *«>ver, after transposition, in Einstem' 
Broad "Eucl^M ^ We " darifled in a " interesting article by I 
• "KM, Newton, and Einstein," Hibbert Journal (April 1920). 


server stationed in the ether, eyeing the beam's passage in that 

21 „ 

medium, believes that the distance covered is really— — . He 

\ c 2 

sees that if the moving clock recorded time like the motionless 


one beside him, it would show an interval Since it 

c \ c 2 

nevertheless shows onlyy, it is because its time is elapsing 
more slowly. If, in the same spatial interval between two events, 
a clock ticks off a fewer number of seconds, each of them lasts 
longer. The second of the clock attached to the moving earth 
is therefore longer than that of the stationary clock in the 
motionless ether. Its duration is —Lp. But the earth-dweller 

is not aware of this. 

More generally, let us again call S a system motionless in the 
ether and S' a double of this system, which at first coincided 
with it and then broke away in a straight line at speed v. As S' 
contracts in the direction of its motion, its time expands. An 
individual attached to system S, perceiving S' and fixing his 
attention upon a clock-second in S' at the exact moment of the 
doubling, would see the second of S growing longer m S like 
an elastic band being stretched, like an arrow seen under a 
magnifying glass. Let us understand: no change has taken place 
in the clock's mechanism or functioning. The phenomenon has 
nothing to do with the lengthening of a pendulum. It is not 
because clocks go more slowly that time has lengthened; it is 
because time has lengthened that clocks, remaining as they are, 
are found to run more slowly. As the result of motion, a longer 
drawn-out, expanded time comes to occupy the spatial interval 
between two positions of the clock hand. The same slowing, 
moreover, obtains for every motion and change m the system, 
since each of them could equally well become representative 
of time and be given the status of a clock. 


We have just been assuming, it is true, that the terrestrial 
observer followed the departure and return of the beam of 
light from 0 to A and from A to O, and measured the speed 
of light without having to consult any other clock than the one 
at point O. What would happen if one were to measure this 
speed only on departure, in that case consulting two clods 
located at points O and A respectively? It is, in truth, the 
beam's double journey that is measured in any terrestrial meas- 
urement of light. The experiment of which we speak has there- 
fore never been performed. But nothing proves it unrealizable. 
We are going to show that it would still give us the same figure 
for the speed of light. But, to that end, let us recall what the 
agreement of our clocks consists of. 

How do we synchronize two clocks located at different 
places? By a communication established between the two in- 
dividuals entrusted with the synchronizing. But, there is no 
instantaneous communication; and, since every transmission 
takes time, we have had to select one that is carried out under 
unchanging conditions. Only signals emitted through the ether 
meet this requirement: all transmission through ponderable 

6 It goes without saying that, in this paragraph, we are giving the name 
of clock to any device allowing us to measure an interval of time or « 
situate two instants in exact relation to one another. In experiments relat- 
ing to the speed of light, Fizeau's cogged wheel and Foucaulfs turning 
mirror are clocks. Still more general will be the meaning of the word * 
*e context of the present study. It will be applied to a natural proc* 
« well. The turning earth will be a clock. 

Moreover, when we speak of the zero of one clock and of the operate" 
by which we determine the zero point of another clock so as to ob*» 
i LTTu 1, h " ° nl ? f0r the sak * °f neater def.niteness that we W* 
n dials and hands. Given any two time-measuring devices whatever nat 

ho 0s ; ;S C,al ; giVen ' consequently, two motion.,, we shall be able t 
Aoo e rburardy any poim Qn ^ moying ^ u* 

»m C o Jy n ^ Ca " h The setting of zero on the second de«« 
A poinri S,mP , ly ° f marki "&. on the path of the second moving ** 

of ze'o" win H 10 ^P 0 " 11 to the »«* "stan, In sh ° rt> thC 3 
0PeraL ,0 * understood, in what follows, as the real or 1*J 

respectrvely. mUUane "y wi » have been marked on the two dev><* 



matter depends upon the state of that matter and the myriad 
circumstances that modify it at every moment. It is therefore 
by means of optical, or, more generally, electromagnetic, sig- 
nals that the two operators have been obliged to communicate 
with each other. The individual at O has dispatched to the 
one at A a beam of light intended to return to him immedi- 
ately. And things have turned out as they did in the Michel- 
son-Morley experiment, with the difference, however, that 
mirrors have been replaced by people. There had been an 
understanding between the two operators at O and A that the 
latter would mark a zero at the point where the hand of his 
clock would be at the precise instant at which the beam would 
reach him. Consequently, the former had only to mark on his 
clock the beginning and end of the time interval taken up by 
the beam's round trip: it is in the middle of this interval that 
he has situated the zero of his clock, since he wished the two 
zeros to mark "simultaneous" moments and the two clocks to 
agree from then on. 

However, this procedure would be perfectly fine only if the 
signal's journey were the same leaving as returning or, in other 
words, if the system to which clocks O and A are attached were 
motionless in the ether. Even in a moving system, it would still 
be fine for the synchronizing of two clocks O and B situated 
on a line perpendicular to the direction of its path; we know, 
in fact, that if the motion of the system leads O to O', the 
beam of light makes the same run from O to B' as from B' to 
O', the triangle OB'O' being isosceles. But it is different in the 
case of the signal's transmission from O to A and vice versa. 
The observer who is at absolute rest in the ether believes that 
the passages are unequal, since in the first journey the beam 
emitted from point O must chase after point A which is fleeing 
it, while on the return trip the beam sent back from point A 
finds point O coming to meet it. Or, if you prefer, he takes 
note that the distance 0,4 /supposedly identical in both cases, 
has been cleared by light at the relative speed of c-v m the 
first, and c + v in the second, so that the times for the distances 
covered are as c + v to c - v. In marking the zero in the middle 


of the interval traversed by the clock hand between the beam's 
departure and return, it is being placed, as our motionless 
observer sees it, too close to the point of departure. Let us cal- 
culate the amount of the error. We said just before that the 
interval traversed by the clock hand on the dial during the 

round trip is — . If, then, at the moment of the signal's emis- 
sion, a provisional zero has been marked at the point where 
the clock hand was, it is at point - of the dial that there will 
have been placed the definitive zero M that corresponds, it is 
believed, to the definitive zero of the clock at A. But the mo- 
tionless observer knows that if the definitive zero of the clod 
at O is really to correspond to that of the clock at A, to be 
simultaneous with it, it would have had to be placed at a point 

that divided the time interval - not into equal parts but into 
parts proportional toc + v and' c-v. Let us call x the first 

of these two parts. We shall have —?L_ = £±H and therefore 

21 c-v 

X J lv T ~* 

c C 2 > which amounts to saying that, for the motionless ob- 
server, the point M where the definitive zero has been marked 
« ? too close to the provisional zero and that, if it is desired to 
eave it where it b, the definitive zero of the clock at A must be 
Pushed back by ^ in order tQ haye a ^ simukaneity between 
the definitive ^zeros of the two clocks. In short, the clock *i 

show^ 5 ^ dial imerVal slower than the time U 0Ught W 
toclil^ d ° Ck hand is at the Po^t that we shall agree 
motion L r nTr'u 116 desi 6™i°n t for the time of the do* 
that if it ' n Cther) ' the motionless observer tells himself 
, 1 V ly a « reed w"h the clock at O, it would sho* 



In that case, what will happen when operators, respectively 
located at O and A, wish to measure the speed of light by 
noting on the synchronized clocks at those points the moment 
of departure, the moment of arrival and, consequently, the 
time that light takes to leap the interval? 

We have just seen that the zeros of the two clocks have been 
so placed that, to anyone considering the clocks as agreeing, 
a light ray always appears to take the same time in going from 
O to A as in returning to it. Our two physicists will therefore 
naturally find that the time for the journey from O to A, com- 
puted by means of the two clocks located at O and A respec- 
tively, is equal to half the round trip's total time, as computed 
on the clock at O alone. But, we know that the duration of 
this round trip, computed on the clock at O, is always the 
same, whatever the speed of the system. It will therefore be so 
again for the duration of the single trip computed by this new 
procedure with two clocks: the constancy of the speed of light 
will again be established. However, the motionless observer in 
the ether will be following what has been happening from 
point to point. He will realize that the distance covered by the 
beam from O to A is proportional to the distance covered from 
A to O in the ratio of c + v to c - v, instead of being equal. He 
will find that, as the zero of the second clock does not agree 
with that of the first, the departure and return times, which 
seem equal when the two clock readings are compared, are 
really as c + » to c-w. There has therefore occurred, he will 
reflect, an error in the length of the distance traveled and an 
error regarding the duration of the journey, but the two errors 
offset each other because it is the same double error that 
earlier presided at the synchronization of the two clocks. 

Thus, whether we compute time on only one clock in a par- 
ticular place or whether we use two clocks at a distance from 
each other, we obtain the same figure for the speed of light 
within the moving system S'. Observers attached to the moving 
system will judge that the second experiment confirms the first. 
But our motionless spectator, based in the ether, will simply 
conclude that he has two corrections to make instead of one 


for everything relating to the time shown by the clocks of sys- 
tern S'. He had already found that these clocks were running 
too slowly. He will now reflect that, in addition, the docb 
ranged along its direction of motion lag behind one another. 
Suppose once more that the moving system S' has been sepa- 
rated, as a double, from the motionless system S, and that the 
dissociation has taken place just as a clock C' 0 in moving sys- 
tem S', coinciding with clock C 0 in system S, pointed, like it, 
to zero. Let us then consider a clock C\ in system S' so placed 
that the straight line C 0 C'\ indicates the direction of the 
system's motion, and let us call / the length of this line. When 
clock C\ shows time V, the motionless observer rightly reflects 
that, since clock C\ has lagged behind clock C 0 of this system 

by a dial interval of ^, there has really elapsed a i' + ^um- 
ber of seconds in system S'. But, having observed the slowing 
lme resul ting from motion, he already knew that each of 
those seeming seconds is equal to 1 of a real second. He 


will therefore calculate that if clock gives a reading of* 
*e time really elapsed is-*, h y Moreover, at thai 

hTwill 1 ^ 11 ^ ° ne of the of his motionless ijH* 

figure 1 thC Ume < which Jt * hows actuall y iS *" 

needed havin S become aware of the con ^ 

£or Z ^ fr ° m time *' to time t, he had perceived * 
in* of siL 1S i C ° mmitted inside the moving system in the jn* 
^S^l H f h3d *»P* " while watching * 
indefini"e] v f 1 d ° Cks - Let us indeed consider ' ° B 1 

ber^ rc" C '° ^ ° f thlS SyStCm ' 3 ^ 
When S' r n - « j »' C * et c, separated by equal intervals o» 

Unless in T Whh S and therefore happened to \** 
,n the ether, the optical signals that came and "«* 



between two successive clocks made equal trips in both direc- 
tions. If all the clocks thus synchronized showed the same time, 
it really was at the same instant. Now that S' has separated 
from S as a result of the doubling, the individual in S', who is 
unaware of being in motion, leaves his clocks C' 0 , C' u C' 2 , etc., 
as they were; he thinks he has real simultaneities when the 
clock hands point to the same dial numeral. Moreover, if he has 
any doubt, he proceeds anew to his synchronizing; he simply 
finds the confirmation of what he had observed in the motion- 
less state. But the motionless onlooker, who sees how the opti- 
cal signal now takes a longer path in going from C' 0 to C\, from 
C\ to C 2 , etc., than in returning from C\ to C' 0 , from C' 2 to 
C\, etc., realizes that to have real simultaneity when the clocks 
show the same time, the zero of clock C\ would have to be 

turned back by^, the zero of clock C 2 by etc. Simulta- 
neity has changed from real to nominal. It has been incurvated 
into succession. 

To sum up, we have just been trying to discover why light 
could have the same speed for both the stationary and the 
moving observer: the investigation of this point has revealed 
that a system S', born of the doubling of a system S, and mov- 
ing in a straight line at a speed v, underwent singular modifi- 
cations. We would formulate them as follows: 

1. All lengths in S' have contracted in the direction of its 
motion. The new length is proportional to the old in the ratio 

of ^1-^ to unity. 

2. The time of the system has expanded. The new sec ond is 

proportional to the old in the ratio of unity to J i _ _ . 

3. What was simultaneity in system S has generally become 
succession in system S'. Only those contemporaneous events in 
S remain contemporaneous in S' which are situated in the same 
plane perpendicular to the S system's direction of motion. Any 


other two events, contemporaneous in S, have separated in S 

by seconds of system S', if by I we mean their distance apart 

computed in the direction of motion of their system, that is, 
the distance between the two planes, perpendicular to this di- 
rection, which pass through each of them respectively. 

In short, considered in space and time, system S' is a double 
of system S which, spatially, has contracted in the direction of 
its motion, and, temporally, expanded each of its seconds; and 
which, finally, has broken up into succession in time every 
simultaneity between two events whose distance apart has nar- 
rowed in space. But these changes escape the observer who is 
part of the moving system. Only the stationary observer is 
aware of them. 

I shall in that case assume that those two well-known ob- 
servers, Peter and Paul, are able to communicate with each 
other. Peter, who knows what has been going on says to Paul- 
The moment you separated from me, your system flattened 
out, your time swelled, your clocks disagreed. Here are the 
correction formulae which will enable you to get back to the 
»uth. It , s up to you to see what you can do with them." »* 
obvjous that Paul would reply: "I shall do nothing, because, 
' \ Used these formulae, everything in my system would, p»* 
ica ly and scientifically, become incoherent. Lengths have 
snmnk say you? But ^ ^ ^ . $ ^ ^ ^ meter & 

liay alongside them; and, as the standard of these lengths * 
sTanZr 15 thdr rdation to the ™*er thus altered, d* 
exp a nai mUS ^ emain What il was - Time, you say further, h- 
S , ft ^ y ° U C ° Um more *an one second while *l 
opt o *h T ° nC? But ' if we assume that S and S' are t* 
aefinuio * 6 P anCt eanh ' the S ' "econd. like that of S, M 
rotat on a a r tain 6xed £ractio " of the planet's period <J 
sam durati "V*" * ou wi » about the* not having 

Sultan laSt «*7 ° nC ^ % 

at poinu c' c"T SUCCCSsions? D ° a11 * ree do ^li 
!• ° 2. C s pomt to the same time when there 



three different moments? But at the different moments at 
which they point to the same time in my system, events occur 
at points C\, C 2 , C' s of my system which were legitimately 
designated contemporaneous in system S; I shall then still 
agree to call them contemporaneous in order not to have to 
take a new view of the relations of these events first among 
themselves, and then with all the others. I shall thereby pre- 
serve all their sequences, relations and explanations. In naming 
as succession what I called simultaneity, I would have an in- 
coherent world or one built on a plan utterly different from 
yours. In this way, all things and all relations among things 
will retain their size, remain within the same frames, come 
under the same laws. I can therefore act as if none of my 
lengths had shrunk, as if my time had not expanded, as if my 
clocks agreed. So much, at least, for ponderable matter, for 
what I carry along with me in the motion of my system; drastic 
changes have occurred in the temporal and spatial relations of 
its parts, but I am not, nor need I be, aware of them. 

"Now, I must add that I regard these changes as fortunate. 
In fact, getting away from ponderable matter, what would not 
my predicament be with regard to light, and, more generally, 
electromagnetic events, had my space and time dimensions re- 
mained as they were! These events are not carried along in the 
motion of my system, not they. It makes no difference that 
light waves and electromagnetic disturbances originate in a 
moving system: the experiment proves that they do not adopt 
its motion. My moving system drops them off on the way, so 
to speak, into the motionless ether, which takes charge of them 
from then on. Even if the ether did not exist, it would be in- 
vented in order to symbolize the experimentally established 
fact of the independence of the speed of light from the motion 
of the source that emitted it. Now, in this ether, before these 
optical facts, in the midst of these electromagnetic events, you 
sit motionless. But I pass through them, and what you perceive 
from your fixed observatory happens to appear quite differ- 
ently to me. The science of electromagnetism, which you have 
so laboriously built up, would have been mine to remake: I 


would have had to modify my once-established equations for 
each new speed in my system. What would I have done in a 
universe so constructed? At the price of what liquidation of all 
science would the soundness of its temporal and spatial rela- 
tions have been bought! But thanks to the contraction of my 
lengths, the expansion of my time, the breakup of my simul- 
taneities, my system becomes, with respect to electromagnetic 
phenomena, the exact imitation of a stationary system. No 
matter how fast it travels alongside a light wave, the latter will 
always maintain the same speed in relation to it, the system will 
be as if motionless with respect to the light wave. All is then 
for the best, and a good genie has arranged things this way. 

"There is, nevertheless, one case in which I shall have to 
take your information into account and modify my measure- 
ments. This is in the matter of framing a unified mathematical 
representation of the universe, that is, of everything happening 
in all the worlds moving with respect to you at every speed. 
In order to establish this representation which would give us, 
once complete and perfect, the relation of everything to every- 
thing else, we shall have to define each point in the universe 
Dy its distances x, y, z f rom three giyen planes at right ang i es , 
av ' „* e 7 11 declar e motionless, and which will intersect on 
axes OX, OY, OZ. Moreover, axes OX, OY, OZ, which will be 
cnosen in preference to all others as the only axes really and 
not conventionally motionless, will be given in your fixed sys- 
reL 1 m ° Ving S y stem in wh ^h I happen to be, I shall 

bom! 7 observatio1 " to axes O'X' , O'Y', O'Z', which are 
toZ£??* t **> and. as I see it, it is by its 

lines th« m the three P lan « intersecting on those 

from vour Y P °/ nt ^ my S ? Stem be defined - SinCC U " 

tatioJof zztv^i of view that the s iobai repre rr; 

my observaS be framed ' 1 must find a wav t0 XtU 

to set u D en * ?° ur ™> OX, OY, OZ, or, in other words, 

be abl^knT • m ^ mCans ° £ wh -« I shall once and for all 

to simplify mattos T Tn 11011 Y ° U W jmt give " ^ 2? 
O'Z' coincided i t haU assume that niy axes O'X', > 
comcided with yours be£ore J rf the t *o 


worlds S and S' (which for the clarity of the present demon- 
stration it will this time be better to make completely different 
from one another), and I shall also assume that OX and, conse- 
quently, O'X' denote the actual direction of motion of S'. This 
being so, it is clear that planes Z'O'X' and X'O'Y' simply glide 
over planes ZOX and XOY respectively, that they ceaselessly 
coincide with them and that consequently y and y' are equal, 
as are z and z'. We are then left to calculate x. If, from the 
moment O' leaves O, I compute a time t' on the clock at point 
x', y', z', I naturally think of the distance from this point to 
plane ZOY as equal to x' + vt'. But in view of the contraction 
to which you call my attention, this length x' + vt' would not 

coincide with your x but with x^T^and consequently what 
you call x is — ■ (x' + vf). This solves the problem. I shall 

not forget, moreover, that the time t', which has elapsed for 
me and which my clock at point x', /, z' shows me, is different 
from yours. When this clock gave me the f reading, the time t 

1 / vx'\ . . 
shown by yours was, as you stated, = yt + -^J. imcn is 

the time t which I shall show you. For time as for space, I shall 
have gone over from my point of view to yours." 

That is how Paul would reply. And he would at the same 
time have laid down the famous "transformation equations" 
of Lorentz, equations which, moreover, if we assume Einstein's 
more general standpoint, do not imply that system S is defi- 
nitely stationary. In fact, we shall soon demonstrate how, after 
Einstein, we can make S any system at all, provisionally im- 
mobilized by the mind, and how it is then necessary to attrib- 
ute to S', considered from the point of view of S, the same 
temporal and spatial distortions that Peter attributed to Paul's 
system. In the hypothesis, hitherto always accepted, of a single 
time and of a space independent of time, it is obvious that if S' 
moves with respect to S at the constant speed v, if x', y', z' are 



the distances from a point M' in the system S' to the three 
planes determined by the three axes O'X', O'Y', O'Z', each at 
right angles to the other two, and if, finally, x, y, z are the 
distances from this same point to the three fixed rectangular 
planes with which the three moving planes were at first merged, 
we have 

x = x' + vt' 


z = z'. 

Moreover, as the same time always unfolds in every system, 
we get 

t = f. 

But, if motion brings about contractions in length, a slowing 
of time, and causes the clocks of the time-expanded system to 
show only a local time, there ensue explanations between Peter 
and Paul until we have 

1 + 

(!) y = y' 

z = z f 


Hence we have a new formula for the composition of speeds. 
Let us, in fact, imagine point M' moving with uniform mo- 
tion inside S', parallel to O'X' at speed x/ measured, of course, 

by-p . What will be its speed for the observer in S who refers 
the successive positions of the moving point to his axes OX, 
OY, OZ? To find this speed v", measured by - , we must divide 
ber, ^ming^ ^ ^ e q u ations member by meffl- 

c 2 



although up till now mechanics laid down that 

v" = v + 1/. 

Accordingly, if S is a river bank and S' a boat sailing at 
speed v with respect to the bank, a passenger walking its deck 
at speed xf in its direction of motion would not have, in the 
eyes of the motionless observer on the shore, speed v + x/, as 
was hitherto believed, but a speed less than the sum of the 
two component speeds. At least, that is how things look at first. 
In reality, the resultant speed is truly the sum of the two com- 
ponent speeds, if the speed of the passenger on the boat is 
measured from the bank, like the speed of the boat itself. Meas- 

x' . 

ured from the boat, the speed xf of the passenger is if, say, 

the length that the passenger finds the boat to be (a constant 
length, since the boat is always at rest for him) is called x' and 
the time he takes to walk it, t', that is, the difference between 
his times of departure and arrival as shown on two clocks 
placed at its stern and bow respectively (we are imagining an 
immensely long boat whose clocks could only have been syn- 
chronized by signals transmitted at a distance). But, for the 
observer motionless on the bank, the boat contracted when it 
passed from rest to motion, time expanded on it, its clocks 
no longer agreed. In his eyes, the distance walked off on the 
boat by the passenger is therefore no longer x' (if x' were the 
length of the quay with which the motionless boat coincided), 

but x'^jl - ^; and the time taken to cover this distance is not 
f but 1 ( t' + — ^ . He will conclude that the speed to be 

added to v in order to get v" is not xf but 


that is, 

He will then have 


. tn/ 

v + x/ 

in/ ~ xnf ' 

1 + ST 1 +- 

C 2 C 2 

We see thereby that no speed can exceed that of light, every 
composition of any speed v' with a speed v assumed equal to c 
always resulting in this same speed c. 

Such are the formulae, therefore-to come back to our first 
nypothesis-which Paul will have in mind when he wishes to 
pass from his point of view to Peter's and thus obtain (every 
observer attached to every moving system S", S'", etc., having 
aone as much) a unified mathematical representation of the 

J? t r r ? 'J f C ° Uld haVC establ ^ed his equations directly, 
widiout Peter . intervention, he could just as well have sup 

S ciuLTx r; n ; r t er T to al,ow him ' knowin ^ - >> z > 

x'- 1 / 


c 2 

equations which are m 
transformation t IW a 6 USUally P resen *d as die Lorentz 
• But this is of small concern at the moment, 
is important to d ' 
Lorentz equations in the" 1 ^ WC have i ust reconstituted ^ 

course of commenting upon the Michelson- 


In rediscovering these equations term by term, in defining the 
perceptions of observers placed in one or the other system, we 
only wished to set the stage for the analysis and demonstration 
that form the subject of the present work. 

Morley experiment, it was with a view to showing the concrete meaning 
of each of the terms that compose them. The fact is that the transforma- 
tion group discovered by Lorentz assures, in a general manner, the invari- 
ance of electromagnetic equations. 


Complete Relativity 

" ing , redpr0dty ° f motion = relativity, no 
first h 7 terar tUt " bilate ^"; interference of this 
W rT, * e SeC ° nd: ensuin S ^understand- 

vefa'nre "* m ° tion; Propagation and con- 

-y-u«, systems of reference; from Descartes to Einstein 

we shall canT m ?. ly Slipped from the P oi "t °* view which 
itTwh ich'sF ° f UnilatCral reI ^-ity" to that of reciproo 

Bmtt t sav in T own - Let us hurr ? back to our p° sidon - 

motion, the exmn that the con ^ctio„ of bodies in 

taneity in^uc^wm ** ^ breakup of simul- 
as they are: ther^n h b \ retamed in Einstein's theory just 
we have just work d nothl ng to change in the equations 
about system S' in't ^ ° r ' m ° re 8 eneraI1 y» in what we said 
tern S. Only these tem ? oral and spatial relations with sys- 
breakups of simuS?" 1 ^ 110115 SizC ' ex P ansions of time - and 
(they are already so ■ beC ° me ex P licitI y reciprocal 

tions), and the observer J/ " the VCrv form of ^ ^ 
observer i n S had assert T t tepeat for 5 everything 46 
as we shall also show h * S '' There wil1 then disa PP ear ' 
°ry of relativity We cl • P aradoxicaI in the the ' 

independent of durat' 3101 ^ * Si " gle time and an extension 
considered in its pureT C ° minue to exis t in Einstein's theory 
been for common se remain what they have always 

^ive at the theory oT " " P racticaI1 y impossible to 
through that of a s i n J * d ° uble relativity without passing 
absolute point of refere tivitv ' wher e one still posits an 
n «, a motionless ether. Even when we 



conceive relativity in the second sense, we still see it a little in 
the first; for say as we will that only the reciprocal motion of 
S and S' with respect to one another exists, we do not investi- 
gate this reciprocity without taking one of the two terms, S or 
S', as our "system of reference"; but, as soon as a system has 
been thus immobilized, it temporarily becomes an absolute 
point of reference, a substitute for the ether. In brief, absolute 
rest, expelled by the understanding, is reinstated by the imagi- 
nation. From the mathematical standpoint, there is no objec- 
tion to this. Whether system S, adopted as a system of refer- 
ence, is at absolute rest in the ether, or whether it is at rest 
solely with respect to every system with which we compare it, 
in both cases the observer located in S will treat alike the 
measurements of time which will be transmitted to him from 
every system such as S'; in both cases, he will apply Lorentz' 
transformation equations to them. The two theories are equiva- 
lent for the mathematician. But the same is not true for the 
philosopher. For if S is at absolute rest and all other systems 
are in absolute motion, the theory of relativity will actually 
imply the existence of multiple times, all on the same footing 
and all real. But if, on the other hand, we subscribe to Ein- 
stein's theory, the multiple times will remain; but there will 
never be more than a single real one among them, as we pro- 
pose to demonstrate; the others will be mathematical fictions. 
That is why, in our opinion, if we adhere strictly to Einstein's 
theory, all the philosophical difficulties relative to time disap- 
pear, and so too will all the oddities that have led so many 
minds astray. We need not, therefore, dwell upon the meaning 
to assign the "distortion of bodies," the "slowing of time," and 
the "rupture of simultaneity" when we believe in the existence 
of a motionless ether and a privileged system. It will be enough 
to try to find out how we ought to understand them in Ein- 
stein's theory. Then, casting a backward glance over the first 
point of view, we shall realize that we had to take that posi- 
tion at first, and we shall consider natural the temptation to 
return to it even though we have adopted the second; but we 
shall also see how false problems arise from the fact alone that 


images have been borrowed from the one to nourish the ab- 
stractions corresponding to the other. 

We have imagined a system S at rest in the motionless ether, 
and a system S' in motion with respect to S. But, the ether has 
never been perceived; it has been introduced into physics as 
a prop for calculations. On the other hand, the motion of a 
system S' with respect to system S is an observed fact. We must 
also consider as a fact, until proven otherwise, the constancy 
of the speed of light in a system that changes speed at our 
bidding and whose speed can therefore drop to zero. Let us 
now return to the three assertions with which we set out: (1) 
S' shifts with respect to S, (2) light has the same speed in both 
systems, (3) S is stationed in a motionless ether. It is clear that 
two of these express facts, and the third, a hypothesis. Let us 
reject the hypothesis: we now have no more than the two facts. 
But, in that case, the first one will no longer be formulated in 
the same way. We stated that S' shifts with respect to S; why 
did we not just as readily declare S to be shifting with respect 
to S'? Simply because S was judged to be sharing the absolute 
immobility of the ether. But there is no longer any ether, 1 no 

1 We are, of course, speaking only of a fixed ether, constituting a privi- 
leged unique, absolute system of reference. But the ether theory, properly 
amended, may very well be picked up again by the theory of relativity. 
Einstein is of this opinion (see his lecture of 1920 on "The Ether and The 
theory of Relativity"). To preserve the ether, the attempt had already 
72 Zf C ? USC S ° me ° £ Larmor ' 5 idea * (ct Ebenezer Cunningham, The 
Pnnaples of Relativity [Cambridge: University Press, 1914], Chap. XV). 
5 low by EinStdn t0 which Ber K*°n refers was delivered on May 

and W p Universi[ y ° f L eyden. It has been translated by G. B. Jeffrey 
flnmT "J 61 ; alon S with 'wo other lectures, in Sidelights on RrWM 
ollows"^ e "' 1922) - In this Einstein sums up his view as 

Tf reTaHvif eCaP " " ting> ™ ^ sa ? that - ^cording to the general theory 
fore tZ Y ' SP3Ce ,S end ° Wed With P h ? si «l qualities. In this sense, there- 
space wTtho ? ^ 11 Cther - AcC ° rdi "S to the general theory of relatW, 
wo d b „„' W ^ Unthink ^le, for, in Lh space, there not only 

for Standards T^ 0 " ° f but aIs ° "° P° ssibilit y of 

fo e *TSL"^: C - ""g -ds Ld clocks), nor there- 

ny space-t,me interval, i„ the physical sense. But this ether m*1 



longer absolute stability anywhere. We shall thereto* .Me 
to say, as we please, that S' is moving with respect to S or that 
S is moving with respect to S>, or rather that S and S are 
loving wil respect to one another. In short, ^ ^ 
given t a reciprocity of displacement. How could it be pother 
lise, since the motion perceived in space „ otd, 'a ^conunual 
variation of distance? If we consider two pomts A and B and 
Z polional change of "one of them," all that the eye per- 
ceives and science can note is the change in the distance be- 
"een them. Language will express this fact in the statement 
that A moves or that B does. It has die choice; but i, ^ would 
be still closer to experience to say that A and B mov ew£ 
respect to one another, or, more simply, that 
between A and B grows shorter or longer. The reapro Uy 
of motion is therefore a fact of observation. We cou d state 
it a priori as a condition of science, because science works 

lengths; and when a ^^^^^Z^ 

reason for privileging one of its extremities,^ an 

is that the distance between the two grows shorter o ^longer 

To be sure it is far from true that every motion is reducible 
to what rpe'rceived of it in space. In addition to motion, > we 
observe only from without, there are also 
scious of producing. When Descartes spoke of the ^city 
of motion" it was not without justice that More replied H 
I am sitting quietly, and someone else, moving a thousand 
paTs aw" g from ml, reddens with fatigue, it is certainly he 

not be thought as endowed with ^^^^^^ 

able media, as consisting of parts winch * ™ g 

The idea o£ motion may not be applied to t (PP- ^J^. it - o£ 

2 We called attention to this point and to that of !*« reap y 
motion in Matiere et ^moire <*^^ I ^ZoLtion to 
1896), Chap. IV; and in Introduction a la ^^'lue \ dg la Morale> 
Metaphysics) (first published in Revue de Metaphystque 

January 1903). mdmoire matter and Memory), pp. 214ft. 

3 On this point see Matiere et memoire (m 
* Descartes, Principles, II, 29. 


who moves and I who rest." 8 All that science can tell us of the 
relativity of the motion perceived by our eyes and measured 
by rulers and clocks will leave untouched our deep-seated feel- 
ing of going through motions and exerting efforts whose dis- 
pensers we are. Let the "quietly seated" More decide to run 
in his turn, let him get up and run: no matter how much we 
insist that his running is a reciprocal place-changing of his 
body and the ground, that he is in motion if our thought im- 
mobilizes the earth, but that it is the earth that moves if we 
consider the runner motionless, he will never accept our rul- 
ing; he will always declare that he perceives his act immedi- 
ately, that this act is a fact, and that the fact is unilateral. All 
men and probably most animals possess this awareness of re- 
solved-upon and executed movements. And since living beings 
Ulus perform motions which really are theirs, which depend 
solely upon them, which are perceived from within but, con- 
smered from without, appear to the eye as nothing more than 
a ^reciprocity of displacement, we can guess that it is so with 
mtive motions generally, and that a reciprocity of displace- 
orr,J S VlSUal manifest ation of an absolute internal change 
in a wlrf in s P a «- We have dwelt upon this point 

seemed t Introduction to Metaphysics. This, in fact, 

penetr 3 tP° US u the function °* the metaphysician: he must 
^derly W n ;°t e T eri ° r of thi ^' «>d the true essence, the 
Wm than I * m ° tl0n can never be better revealed 10 

doubtless «ni P erforms the motion himself, when he 

<ion, bu n IE 6 1VCS U fr ° m the outside like an ? other ** 
whose trart i a PP rehe nds it from within as an effort 

this direct in ^ Visible " But ' the metaphysician obtains 
he himself neT' ^ P erce P tion only from motions that 
absolute motionT^ 0 " 17 these can he guarantee as real acts, 
other living ere t • in Case of motions performed by 
but by symoarh k S ' " " n ° l b ? virtue of a direct perception 
status of indent ° f analo gy that he g ives them 

of matter in ee ni if realities - A nd concerning the motions 
sg w 8 31 ne can sa Y nothing except that there prob- 
M ° re " S ^ ta p ^ ophica (1679) , „ g 24g 



ably are internal changes, analogous or not to efforts, which 
occur we know not where and which are brought before our 
eyes, like our own acts, by the reciprocal displacement of 
bodies in space. We do not therefore have to take absolute 
motion into account in the construction of science; we know 
only rarely where it occurs, and, even then, science would have 
nothing to do with it, for it is not measurable and the business 
of science is to measure. Science can and must retain of reality 
what is spread out in homogeneous, measurable space. The 
motion it studies is therefore always relative and can only con- 
sist of a reciprocity of displacement. Whereas More spoke as 
a metaphysician, Descartes indicated the point of view of sci- 
ence with lasting precision. He even went well beyond the 
science of his day, beyond Newtonian mechanics, beyond our 
own, formulating a principle whose demonstration was re- 
served for Einstein. 

For it is a remarkable fact that the radical relativity of 
motion, postulated by Descartes, could not be categorically 
asserted by modern science. Science, as we understand it since 
Galileo, undoubtedly believed motion to be relative. It gladly 
declared it so. But as a consequence it dealt with it hesitantly 
and incompletely. There were two reasons for this. First, sci- 
ence runs counter to common sense only when strictly neces- 
sary. So, if every rectilinear and nonaccelerated motion is 
clearly relative, if, therefore, in the eyes of science, the track 
is as much in motion with respect to the train as the train is 
with respect to the track, the scientist nonetheless declares that 
the track is motionless; he speaks like anyone else when he has 
no interest in expressing himself otherwise. But this is not the 
main point. The reason that science has never insisted upon 
the radical relativity of uniform motion is that it felt incapa- 
ble of extending this relativity to accelerated motion— at least 
it was obliged to give up the attempt provisionally. More than 
once in the course of its history it has submitted to a necessity 
of this sort. From a principle immanent in its method, it sacri- 
fices something to an hypothesis which is immediately verifi- 
able and which gives useful results right away. If the advantage 


continues, it is because the hypothesis was true in one respect; 
and consequently, this hypothesis will perhaps one day be 
found to have definitely contributed to establishing the prin- 
ciple that it had provisionally set aside. It is thus that New- 
tonian dynamics appeared to cut short the development of 
Cartesian mechanics. Descartes posited that everything relating 
to physics is spread out and moving in space; he thereby gave 
the ideal formula of a universal mechanism. But, to cling to 
this formula would have meant considering globally the rela- 
tion of all to all; whereas a solution, albeit provisional, of 
particular problems could be obtained only by more or less 
artificially carving out and isolating parts within the whole; 
but, as soon as relation is neglected, force is introduced. This 
introduction was only that very elimination; it expressed the 
necessity, under which the human intellect labors, of studying 
reality a portion at a time, powerless as it is to form, at one 
stroke, a combined analytic and synthetic conception of the 
whole. Newton's dynamics could therefore be-and has indeed 
turned out to be-a step toward the complete demonstration 
of Cartesian mechanics, which Einstein has perhaps achieved. 
But, this dynamics implied the existence of an absolute mo- 
tion. One could still grant the relativity of motion for non- 
accelerated rectilinear translation; but the appearing of cen- 
trifugal forces in rotational motion seemed to attest that one 
was now dealing with a true absolute; and that all other 
accelerated motion was equally to be considered absolute. Such 
is the theory that remained classic until Einstein. It was, how- 
ever, not possible to get more than a provisional understand- 
ing from it. A historian of mechanics, Mach, had drawn atten- 
tion to its inadequacy,* and his critique certainly helped giv e 
rise to new ideas. No philosopher could be entirely satisfied 
with a theory that regarded mobility as an ordinary relation 
of reciprocity in the case of uniform motion, and as a reality 
immanent m a moving body in the case of accelerated motion- 
it, for our part, we thought it necessary to admit of an absolute 
change wherever a spatial motion is observed, if we believed 

• Ernst Mach, Die Mechanik in ihrer Entwicklung, II, vi. 



that the consciousness of effort reveals the absolute character 
of the attendant motion, we added that the consideration of 
this absolute motion concerns only our knowledge of the inte- 
rior of things, that is, a psychology that reaches into meta- 
physics. 7 We added that, for physics, whose role is to study 
the relations among visual data in a homogeneous space, every 
motion had to be relative. And yet certain motions could not 
be so. Today they can. If only for this reason, the general 
theory of relativity marks an important date in the history of 
ideas. We do not know what final fate physics reserves for it. 
But, whatever happens, the conception of spatial motion which 
we find in Descartes, and which harmonizes so well with the 
spirit of modern science, has been rendered scientifically ac- 
ceptable by Einstein for accelerated as for uniform motion. 

It is true that this part of Einstein's work is the last. It is 
the "generalized" theory of relativity. The reflections upon 
time and simultaneity belong to the "special" theory of rela- 
tivity, the latter being concerned only with uniform motion. 
But within the special theory there was a kind of demand for 
the general theory. For despite its being "special," that is, 
limited to uniform motion, it was not the less radical, since it 
declared motion to be reciprocal. Now, why had one not yet 
gone that far openly? Why was the idea of relativity applied 
only hesitantly even to the uniform motion that was declared 
relative? Because it was feared that the idea would no longer 
apply to accelerated motion. But, as soon as a physicist regards 
the relativity of motion as radical, he has to try to envisage 
accelerated motion as relative. Were it still only for this reason, 
the special theory of relativity drew in its wake that of general 
relativity and could appear convincing to the philosopher only 
by lending itself to this generalization. 

But if all motion is relative and if there is no absolute point 
of reference, no privileged system, the observer inside a system 
will obviously have no way of knowing whether his system is 
in motion or at rest. Nay, let us say that he would be wrong 

7 Matiire et memoire {Matter and Memory), 214ft. Cf. Introduction a la 
me'taphysique (Introduction to Metaphysics). 


to wonder about it, for the question no longer has any mean- 
ing; it does not present itself in those terms. He is free to rule 
whatever he pleases; his system will be motionless, by very 
definition, if he makes it his "system of reference" and there 
installs his observatory. This could not be so even in the case 
of uniform motion when we believed in a motionless ether; it 
certainly could not be so when we believed in the absolute 
character of accelerated motion. But as soon as these two 
theories are discarded, any system is at rest or in motion, as 
we please. It is, of course, necessary to abide by the choice 
of the motionless system once made and to treat the others 

We do not wish to prolong this introduction unduly. We 
must nevertheless recall what we once said about the idea of 
body and also of absolute motion; that double series of reflec- 
tions permitted us to infer the radical relativity of motion as 
displacement in space. What is immediately given to our per- 
ception, we explained, is a continuity of extension upon which 
qualities are deployed; more especially, it is a visual continuity 
of extension, and, therefore, of color. There is nothing here of 
the artificial, conventional, merely human. Colors would prob- 
ably appear differently to us if our eye and our consciousness 
were differently formed; nonetheless there would always be 
something unshakably real which physics would continue to 
resolve into elementary vibrations. In brief, as long as we speak 
only of a qualified and qualitatively modified continuity, such 
as colored and color-changing extension, we immediately 
express what we perceive, without interposed human conven- 
tion-we have no reason to suppose that we are not here in 
^e presence 0 f reality itself. Every appearance must be deemed 
a reality as long as it has not been shown to be illusory, and 

wa tv, TTT ^ been made ^ ^ actual case; it 

>u- i e been made but that was an illusion, as 

present ^ P roven - 8 Matter is therefore immediately 

ST£ l a rea lty - But is this true for a p articuiar body 

given the status of a more or less independent reality? The 

^ Ct ^ m0 ' Ve < M *»- Memory), pp. 225ff . C f. Chap. L 



visual perception of a body is the result of our dividing up of 
colored extension; we have cut it out of the continuity of ex- 
tension. It is very likely that this fragmentation is earned out 
differently by different animal species. Many are incapable of 
aoing ahead with it; and those who are able to are governed, 
in this operation, by their type of activity and the nature of 
their needs. "Bodies," we wrote, "have been cut out of nature s 
cloth by a perception whose scissors follow the stippled lines 
over which action would pass." 9 That is what psychological 
analysis has to say. And physics confirms it. It dissolves the 
body into a virtually infinite number of elementary corpuscles; 
and, at the same time, it shows us this body linked to other 
bodies by thousands of reciprocal actions and reactions. It thus 
introduces so much discontinuity into it, and, on the other 
hand, establishes between it and the rest of things so much 
continuity that we can gather what there must be of the arti- 
ficial and conventional in our division of matter into bodies. 
But, if each body, taken individually and arrested where our 
habits of perception bound it, is in great part a being of con- 
vention, why would this not be so for the motion considered 
to be affecting this body individually? There is only one mo- 
tion, we said, which is perceived from within, and of which we 
are aware as an event in itself: the motion that our effort 
brings to our attention. Elsewhere, when we see a motion 
occur, all we are sure of is that some change is taking place in 
the universe. The nature and even the exact location of this 
change escape us; we can only note certain changes of position 
that are its visual and surface aspect, and these changes are 
necessarily reciprocal. All motion-even ours as perceived from 
without and made visual-is therefore relative. It goes without 
saying, moreover, that only the motion of ponderable matter 
is in question. The analysis just made shows this clearly 
enough. If color is a reality, so must be the oscillations that 
somehow occur within it-since they have an absolute charac- 

*L'Evolution criatrice (Creative Evolution) (Paris: F. Alcan, 1907), pp. 
12, 13. Cf. Maliere et memoire {Matter and Memory), Chap. I ana pp. 


ter, ought we still to call them motions? Furthermore, how can 
we rank the act by which these real oscillations, elements of a 
quality and partaking of what is absolute in the quality, are 
propagated in space with the entirely relative, necessarily re- 
ciprocal displacement of two systems S and S' carved more or 
less artifiaally out of matter? We speak of motion, here as 
there; but has the word the same meaning in both cases? Let 
us rather speak of propagation in the first and conveyance in 
the second: lt follows from our analyses of old that propaga- 
ion must be thoroughly distinguished from conveyance. But, 
tnen rejecting the emission theory, the propagation of light 

Te^7v\ tT ^ ti0n ° f P artides ' we not expect die 

Zthl 1 f ght l With res P ect *> a system to vary in accordance 
k til T 6 Ia " er " rCSt " or " in m ^ion." Why should 

ing ^ ^sZ^ Uin CntireIy hUman W ^ ° f PerCCiV ' 

not W^dSin "a 6 ^ ^ We ** 

meanintr w I u gCneral man ner certain terms whose 

^ ^"e b;te^r ed ****** * "* 

we shall apply 2^Z > W made ° f them - Adding!* 
trirectanrie wIk SyStem of reference" to the trihedral 

indicating theh^ £££ * ^ W sha11 **™ to ^ * 

Points i/thetnSe Th T-^ b ° m itS ^ ^ 
will be attarh^ , u P h ysicist who is building Science 

will serve ht 1 *!? The Vertex <> f this "rihedral 

system of referenced [ "* ^ ° bservator y- The points of his 
one another Bm cou rse, be at rest with respect to 

relativity, the system^ I* ^ that * in the h yP othesis ° f 
toe while it is beinT V f eference w iH itself be motionless all 
fixi ty of a trihedral referring. What, in effect, can the 

npon it, the momentaril^- ^ if " 0t 3 P ro P ert Y we bestoW 
it, in adopting it , 7 P nvue ged situation that we assure 

f ull s it as our svct^tv. „r r 
retain a stationary ether ? . reference? As long as we 
belongs to things in e abs °lute positions, immobility 

the ether has vanishe^T^ " " not of our decreeing. Once 
wished along with the privileged system and 


fixed points, only relative motions of objects with respect to 
one another are left; but, as we cannot move with respect to 
ourselves, immobility will be, by definition, the state of the 
observatory in which we shall mentally take our place: there, 
as a matter of fact, is our trihedral of reference. To be sure, 
nothing prevents us from imagining, at a given moment, that 
the system of reference is itself in motion. Physics is often inter- 
ested in doing so, and the theory of relativity readily makes 
this assumption. But when the physicist sets his system of refer- 
ence in motion, it is because he provisionally chooses another, 
which then becomes motionless. It is true that this second sys- 
tem can in turn be mentally set in motion without thought 
necessarily electing to settle in a third system. But in that case 
it oscillates between the two, immobilizing them by turns 
through goings and comings so rapid that it entertains the illu- 
sion of leaving them both in motion. It is in this precise sense 
that we shall speak of a "system of reference." 

On the other hand, we shall apply the term "constant sys- 
tem" or simply "system," to every group of points which retain 
the same relative positions and which are therefore motionless 
with respect to one another. The earth is a system. A multi- 
tude of displacements and changes no doubt appear on its 
surface and hide within it; but these motions stay within a 
fixed frame; I mean that no matter how many relatively fixed 
points we find on earth we cannot help but be attached to 
them, the events that unfold in the intervals then passing as 
mere mental views: the events would be nothing more than 
images successively combing through the consciousness of mo- 
tionless observers at those fixed points. 

Now a "system" can generally be given the status of a "sys- 
tem of reference." It will be necessary to understand by this 
that we are agreeing to settle the chosen system of reference 
in this system. It will sometimes be necessary to indicate the 
particular point in the system at which we are locating the 
vertex of the trihedral. More often, this will be unnecessary. 
Thus, when we shall be taking account only of the state of rest 
or motion of the system earth with respect to another system, 


it will be possible to view it as a single physical point; this 
point will then become the vertex of our trihedral. Or else, 
allowing the earth its true size, we shall understand that the 
trihedral is located somewhere upon it. 

Moreover, the transition from "system" to "system of refer- 
ence" will be continuous if we take the position of the theory 
of relativity. It is, in fact, essential in this theory to disperse 
an endless number of synchronized clocks, and therefore ob- 
servers, over its "system of reference." The system of reference 
can therefore no longer be a single trihedral with a single 
observer. "Clocks" and "observers" need not be anything physi- 
cal; by "clock" we simply mean here an ideal recording of time 
according to definite laws or rules, and by "observer," an ideal 
reader of this ideally recorded time. It is nonetheless true that 
we are now picturing the possibility of physical clocks and 
living observers at every point in the system. The tendency not 
to differentiate between "system" and "system of reference' 
was, moreover, immanent in the theory of relativity from the 
beginning, since it was by immobilizing the earth, by taking 
this composite system as our system of reference, that the in- 
variability of the result of the Michelson-Morley experiment 
was explained. In most cases, the assimilation of the system of 
reference to an aggregate system of this type offers no objec- 
tion. And it may have great advantages for a philosopher who 
is trying to find out, for example, in what measure Einstein's 
times are real times, and who will therefore be obliged to post 
flesh-and-blood observers, conscious beings, at all the points in 
the system of reference where there are "clocks." 

Such are the preliminary thoughts that we needed to pre- 
sent. We have given them much space. But it was for not 
having strictly defined the terms used, for not having been 
sufficiently used to seeing a reciprocity in relativity, for not 
having constantly borne in mind the relation between the radi- 
cal and the less thoroughgoing relativity, and for not having 
been on our guard against a confusion between them, in f 
word, for not having kept close to the passage from the phy* 


cal to the mathematical that we have been so seriously mis- 
taken about the philosophical meaning of time in the theory 
of relativity. Let us add that we have hardly any longer been 
preoccupied with the nature of time itself. Nevertheless we 
had to begin this way. Let us pause at this point. The analyses 
and distinctions that we have just made, and the reflections on 
time and its measurement that we are about to present will 
make it easy to deal with the interpretation of Einstein s 


Concerning the Nature of Time 

Succession and consciousness; origin of the idea of a uni- 
versal time; real duration and measurable time; concern- 
ing the immediately perceived simultaneity: simultaneity 
of flow and of the instant; concerning the simultaneity 
indicated by clocks; unfolding time; unfolding time and 
the fourth dimension; how to recognize real time 

There is no doubt but that for us time is at first identical with 
the continuity of our inner life. What is this continuity? That 
of a flow or passage, but a self-sufficient flow or passage, the 
flow not implying a thing that flows, and the passing not pre- 
supposing states through which we pass; the thing and the 
state are only artificially taken snapshots of the transition; and 
this transition, all that is naturally experienced, is duration 
itself. It is memory, but not personal memory, external to what 
it retains, distinct from a past whose preservation it assures; it 
is a memory within change itself, a memory that prolongs the 
before into the after, keeping them from being mere snap- 
shots appearing and disappearing in a present ceaselessly re- 
born. A melody to which we listen with our eyes closed, heed- 
ing it alone, comes close to coinciding with this time which is 
the very fluidity of our inner life; but it still has too many 
qualities, too much definition, and we must first efface the dif- 
ference among the sounds, then do away with the distinctive 
features of sound itself, retaining of it only the continuation 
ot what precedes into what follows and the uninterrupted 
transition, multiplicity without divisibility and succession with- 
out separation, in order finally to rediscover basic time. Such 



is immediately perceived duration, without which we would 

have no idea of time. 

How do we pass from this inner time to the time of things? 
We perceive the physical world and this perception appears, 
rightly or wrongly, to be inside and outside us at one and the 
same time; in one way, it is a state of consciousness; in another, 
a surface film of matter in which perceiver and perceived coin- 
cide. To each moment of our inner life there thus corresponds 
a moment of our body and of all environing matter that is 
"simultaneous" with it; this matter then seems to participate 
in our conscious duration.* Gradually, we extend this duration 
to the whole physical world, because we see no reason to limit 
it to the immediate vicinity of our body. The universe seems 
to us to form a single whole; and, if the part that is around «s 
endures in our manner, the same must hold, we think for that 
part by which it, in turn, is surrounded, and so on indefinitely. 
Thus is born the idea of a duration of the universe, that is to 
say, of an impersonal consciousness that is the link among all 
individual consciousnesses, as between these consciousnesses 
and the rest of nature.* Such a consciousness would grasp, in a 
single, instantaneous perception, multiple events lying at dil- 
ferent points in space; simultaneity would be precisely the 
possibility of two or more events entering within a sing e 
instantaneous perception. What is true and what 
this way of seeing things? What matters at the moment is not 
allotting it shares of truth or error but seeing Nearly where ex- 
perience ends and theory begins. There is no doubt that our 
consciousness feels itself enduring, that our perception plays 

XFor the development of the views presented here, see Essai «r Us 
donnees immediate* de la conscience (Time and Free ^ < P * 
Alan. 1889). mainly Chaps. II and III; Matiere et men ^^Jon), 
Memory), Chaps I and IV; VEvolution creatnce (Cr ea ,« o) 
passim. CI. Introduction a la metaphysique C^^.^gS 
and La perception du element (The ^^J^££ in 
Oxford University Press, 1911). [The last-named Utle was rep 
Paris in 1934, along with several other essays, under the Utle L p 
et le mouvant and was translated as The Creative Mind.] 
2 Cf. those of our works we have just cited. 



part in our consciousness, and that something of our body and 
environing matter enters into our perception. 3 Thus, our dura- 
tion and a certain felt, lived participation of our physical sur- 
roundings in this inner duration are facts of experience. But, 
m the first place, the nature of this participation is unknown, 
as we once demonstrated; it may relate to a property that 
things outside us have, without themselves enduring, of mani- 
festing themselves in our duration in so far as they act upon 
us, and of thus scanning or staking out the course of our con- 
scious life." Next, in assuming that this environment "endures," 
there is no strict proof that we may find the same duration 
again when we change our surroundings; different durations, 
differently rhythmed, might coexist. We once advanced a the- 
ory of that kind with regard to living species. We distinguished 
durations of higher and lower tension, characteristic of differ- 
ent levels of consciousness, ranging over the animal kingdom, 
anu, we did not perceive then, nor do we see even today any 
reason for extending this theory of a multiplicity of durations 
to the physical universe. We had left open the question of 
world CT OT Universe was divisible into independent 

the J^ We , WerC sufficientl y occupied with our own world and 
To deriH IT mpetUS that life manifests there. But if we had 

^i2e tT2eT' w t would ' in our p resent state o£ 

and univeml Th hy P° thesis of a P h r^al time that is one 
an armZTK W Y a h yP oth «is, but it is based upon 

long as we IrJ^T WC muSt re S ard as <™ dusiv£ * 
this scarrll d n ° thin S more satisfactory. We believe 

human co t;° nSaOUS "P™" re ^es to the following: All 
^y keeTir, f USneSSeS m ° f like nature « P er ceive in the same 
nothing prevent' V' ^ and live the ™e duration. But, 
nesses as we niLZ ? ! ma & inin g *s many human conscious- 
verse, but broult , 7 scattere d through the whole uW 
consecutive onef, v Cn ° Ugh to one another for an ? ^ 
'See Mature e " rand ° m ' * ^ ^ 

*CL Essai sur lrZT 6 ^ ^ Memory), Chap. I. 
W 'lt), especially pp. 82B ,mm ^iates de la conscience (Time and Fr« 



their fields of outer experience. Each of these two outer experi- 
ences participates in the duration of each of the two conscious- 
nesses. And, since the two consciousnesses have the same rhythm 
of duration, so must the two experiences. But the two experi- 
ences have a part in common. Through this connecting link, 
then, they are reunited in a single experience, unfolding in a 
single duration which will be, at will, that of either of the two 
consciousnesses. Since the same argument can be repeated step 
by step, a single duration will gather up the events of the whole 
physical world along its way; and we shall then be able to 
eliminate the human consciousnesses that we had at first laid 
out at wide intervals like so many relays for the motion of our 
thought; there will be nothing more than an impersonal time 
in which all things will pass. In thus formulating humanity's 
belief, we are perhaps putting more precision into it than is 
proper. Each of us is generally content with indefinitely en- 
larging, by a vague effort of imagination, his immediate physi- 
cal environment, which, being perceived by him, participates 
in the duration of his consciousness. But as soon as this effort 
is precisely stated, as soon as we seek to justify it, we catch 
ourselves doubling and multiplying our consciousness, trans- 
porting it to the extreme limits of our outer experience, then, 
to the edge of the new field of experience that it has thus dis- 
closed, and so on indefinitely-they are really multiple con- 
sciousnesses sprung from ours, similar to ours, which we en- 
trust with forging a chain across the immensity of the universe 
and with attesting, through the identity of their inner dura- 
tions and the contiguity of their outer experiences, the single- 
ness of an impersonal time. Such is the hypothesis of common 
sense. We maintain that it could as readily be considered Ein- 
stein's and that the theory of relativity was, if anything, meant 
to bear out the idea of a time common to all things. This idea, 
hypothetical in any case, even appears to us to take on special 
rigor and consistency in the theory of relativity, correctly 
understood. Such is the conclusion that will emerge from our 
work of analysis. But that is not the important point at the 
moment. Let us put aside the question of a single time. What 



we wish to establish is that we cannot speak of a reality that 
endures without inserting consciousness into it. The metaphy- 
sician will have a universal consciousness intervene directly. 
Common sense will vaguely ponder it. The mathematician, it 
is true, will not have to occupy himself with it, since he is 
concerned with the measurement of things, not their nature. 
But if he were to wonder what he was measuring, if he were 
to fix his attention upon time itself, he would necessarily pic- 
ture succession, and therefore a before and after, and conse- 
quently a bridge between the two (otherwise, there would be 
only one of the two, a mere snapshot); but, once again, it is 
impossible to imagine or conceive a connecting link between 
the before and after without an element of memory and, conse- 
quently, of consciousness. 

We may perhaps feel averse to the use of the word "con- 
sciousness" if an anthropomorphic sense is attached to it- 
But to imagine a thing that endures, there is no need to take 
one's own memory and transport it, even attenuated, into the 
interior of the thing. However much we may reduce the 
intensity of our memory, we risk leaving in it some degree 
of the variety and richness of our inner life; we are then 
preserving the personal, at all events, human character of 
memory. It 1S the opposite course we must follow. We shall 
nave to consider a moment in the unfolding of the universe, 
wat is, a snapshot that exists independently of any con- 
^lousness, then we shall try conjointly to summon another 
moment brought as close as possible to the first, and thus have 
fZTTT ° f Ume Cnter into the world without allowing the 
tamtest ghmmer of memory to go with it. We shall see that 

nects HIT With ° Ut an ^mentary memory that con- 
conseouen T m ° mentS ' wil1 be onl Y one or the other, 
sfon no d Y w lngle inStam ' no bef <> r e ™1 after, no succes- 
neeriVH t« "^i , C3n bestow u P on this memory just what i» 
connect on COnnecti °n; it will be, if we like, this very 

-dTaHter "h * ** **** * 

what is nnt I Perpetutally renewed forgetfulness of 

what is not the immediately p rior 'moment. We shall nonethe- 


less have introduced memory. To tell the truth, it is impos- 
sible to distinguish between the duration, however short it 
may be, that separates two instants and a memory that con- 
nects them, because duration is essentially a continuation 
of what no longer exists into what does exist. This is real 
time, perceived and lived. This is also any conceived time, 
because we cannot conceive a time without imagining it as 
perceived and lived. Duration therefore implies consciousness; 
and we place consciousness at the heart of things for the very 
reason that we credit them with a time that endures. 

However, the time that endures is not measurable, whether 
we think of it as within us or imagine it outside of us. Meas- 
urement that is not merely conventional implies, in effect, 
division and superimposition. But we cannot superimpose 
successive durations to test whether they are equal or unequal; 
by hypothesis, the one no longer exists when the other appears; 
the idea of verifiable equality loses all meaning here. More- 
over, if real duration becomes divisible, as we shall see, by 
means of the community that is established between it and 
the line symbolizing it, it consists in itself of an indivisible and 
total progress. Listen to a melody with your eyes closed, 
thinking of it alone, no longer juxtaposing on paper or an 
imaginary keyboard notes which you thus preserved one for 
the other, which then agreed to become simultaneous and 
renounced their fluid continuity in time to congeal in space; 
you will rediscover, undivided and indivisible, the melody 
or portion of the melody that you will have replaced within 
pure duration. Now, our inner duration, considered from 
the first to the last moment of our conscious life, is something 
like this melody. Our attention may turn away from it and, 
consequently, from its indivisibility; but when we try to cut 
it, it is as if we suddenly passed a blade through a flame-we 
divide only the space it occupied. When we witness a very 
rapid motion, like that of a shooting star, we quite clear y 
distinguish its fiery line divisible at will, from the indivisible 
mobility that it subtends; it is this mobility that is pure dura- 
tion. Impersonal and universal time, if it exists, is m vain 


endlessly prolonged from past to future; it is all of a piece; 
the parts we single out in it are merely those of a space that 
delineates its track and becomes its equivalent in our eyes; 
we are dividing the unfolded, not the unfolding. How do we 
first pass from the unfolding to the unfolded, from pure 
duration to measurable time? It is easy to reconstruct the 
mechanism of this operation. 

If I draw my finger across a sheet of paper without looking 
at it, the motion I perform is, perceived from within, a con- 
tinuity of consciousness, something of my own flow, in a word, 
duration. If I now open my eyes, I see that my finger is tracing 
on the sheet of paper a line that is preserved, where all is 
juxtaposition and no longer succession; this is the unfolded, 
which is the record of the result of motion, and which will 
be its symbol as well. Now, this line is divisible, measurable. 
In dividing and measuring it, I can then say, if it suits me, 
that I am dividing and measuring the duration of the motion 
that is tracing it out. 

It is therefore quite true that time is measured through the 
intermediary of motion. But it is necessary to add that, if this 
measurement of time by motion is possible, it is, above all, 
because we are capable of performing motions ourselves and 
because these motions then have a dual aspect. As muscular 
sensation, they are a part of the stream of our conscious life, 
they endure; as visual perception, they describe a trajectory, 
they claim a space. I say "above all" because we could, i» 
a pinch, conceive of a conscious creature reduced to visual 
perception who would yet succeed in framing the idea of meas- 
urable time. Its life would then have to be spent in the 
contemplation of an outside motion continuing without end. 
It would also have to be able to extract from the motion per' 
ceived in space and sharing the divisibility of its trajectory, 
the "pure mobility," the uninterrupted solidarity of the before 
and after that is given in consciousness as an indivisible fact- 
We drew this distinction just before when we were speaking 
of the fiery path traced out by the shooting star. Such a con- 
sciousness would have a continuity of life constituted by the 



uninterrupted sensation of an external, endlessly unfolding 
mobility. And the uninterruption of unfolding would still 
remain distinct from the divisible track left in space, which 
is still of the unfolded. The latter is divisible and measurable 
because it is space. The other is duration. Without the con- 
tinual unfolding, there would be only space, and a space that, 
no longer subtending a duration, would no longer represent 

Now, nothing prevents us from assuming that each of us 
is tracing an uninterrupted motion in space from the begin- 
ning to the end of his conscious life. We could be walking day 
and night. We would thus complete a journey coextensive 
with our conscious life. Our entire history would then unfold 
in a measurable time. 

Are we thinking of such a journey when we speak of an 
impersonal time? Not entirely, for we live a social and even 
cosmic life. Quite naturally we substitute any other person's 
journey for the one we would make, then any uninterrupted 
motion that would be contemporaneous with it. I call two 
flows "contemporaneous" when they are equally one or two 
for my consciousness, the latter perceiving them together as 
a single flowing if it sees fit to engage in an undivided act 
of attention, and, on the other hand, separating them through- 
out if it prefers to divide its attention between them, even 
doing both at one and the some time if it decides to divide 
its attention and yet not cut it in two. I call two instantaneous 
perceptions "simultaneous" that are apprehended in one and 
the same mental act, the attention here again being able to 
make one or two out of them at will. This granted, it is easy 
to see that it is entirely in our interest to take for the "unfold- 
ing of time" a motion independent of that of our own body. 
In truth, we find it already taken. Society has adopted it for us. 
It is the earth's rotational motion. But if we accept it, if we 
understand it as time and not just space, it is because a 
journey of our own body is always virtual in it, and could 
have been for us the unfolding of time. 

It matters little, moreover, what moving body we adopt as 


our recorder of time. Once we have exteriorized our own dura- 
tion as motion in space, the rest follows. Thenceforth, time 
will seem to us like the unwinding of a thread, that is, like 
the journey of the mobile entrusted with computing it. We 
shall say that we have measured the time of this unwinding 
and, consequently, that of the universal unwinding as well. 

But all things would not seem to us to be unwinding along 
with the thread, each actual moment of the universe would 
not be for us the tip of the thread, if we did not have the con- 
cept of simultaneity at our disposal. We shall soon see the role 
of this concept in Einstein's theory. For the time being, we 
would like to make clear its psychological origin, about which 
we have already said something. The theoreticians of relativity 
never mention any simultaneity but that of two instants. 
Anterior to that one, however, is another, the idea of which is 
more natural: the simultaneity of two flows. We stated that 
it is of the very essence of our attention to be able to be 
divided without being split up. When we are seated on the 
bank of a river, the flowing of the water, the gliding of a boat 
or the flight of a bird, the ceaseless murmur in our life's deeps 
are for us three separate things or only one, as we choose. We 
can interiorize the whole, dealing with a single perception that 
carries along the three flows, mingled, in its course; or we can 
leave the first two outside and then divide our attention be- 
tween the inner and the outer; or, better yet, we can do both 
at one and the same time, our attention uniting and yet dif- 
ferentiating the three flows, thanks to its singular privilege of 
being one and several. Such is our primary idea of simultaneity- 
We therefore call two external flows that occupy the same 
duration "simultaneous" because they both depend upon the 
duration of a like third, our own; this duration is ours only 
when our consciousness is concerned with us alone, but it 
becomes equally theirs when our attention embraces the three 
flows in a single indivisible act. 

Now from the simultaneity of two flows, we would never 
pass to that of two instants, if we remained within pure dura- 
tion, for every duration is thick; real time has no instants. 



But we naturally form the idea of instant, as well as of 
simultaneous instants, as soon as we acquire the habit of con- 
verting time into space. For, if a duration has no instants, 
a line terminates in points. 5 And, as soon as we make a line 
correspond to a duration, to portions of this line there must 
correspond "portions of duration" and to an extremity of the 
line, an "extremity of duration"; such is the instant— some- 
thing that does not exist actually, but virtually. The instant 
is what would terminate a duration if the latter came to a halt. 
But it does not halt. Real time cannot therefore supply the 
instant; the latter is born of the mathematical point, that 
is to say, of space. And yet, without real time, the point would 
be only a point, not an instant. Instantaneity thus involves two 
things, a continuity of real time, that is, duration, and a 
spatialized time, that is, a line which, described by a motion, 
has thereby become symbolic of time. This spatialized time, 
which admits of points, ricochets onto real time and there gives 
rise to the instant. This would not be possible without the 
tendency— fertile in illusions— which leads us to apply the mo- 
tion against the distance traveled, to make the trajectory coin- 
cide with the journey, and then to decompose the motion over 
the line as we decompose the line itself; if it has suited us to 
single out points on the line, these points will then become 
positions" of the moving body (as if the latter, moving, could 
ever coincide with something at rest, as if it would not thus 
stop moving at oncel). Then, having dotted the path of motion 
with positions, that is, with the extremities of the subdivisions 
°f the line, we have them correspond to "instants" of the 
continuity of the motion— mere virtual stops, purely mental 
views. We once described the mechanism of this process; we 
have also shown how the difficulties raised by philosophers 
over the question of motion vanish as soon as we perceive the 
relation of the instant to spatialized time, and that of spatial- 

6 That the concept of the mathematical point is natural is well known 
to those who have taught geometry to children. Minds most refractory to 
the first elements imagine immediately and without difficulty lines without 
thickness and points without size. 


ized time to pure duration. Let us confine ourselves here to 
remarking that no matter how much this operation appears 
learned, it is native to the human mind; we practice it instinc- 
tively. Its recipe is deposited in the language. 

Simultaneity of the instant and simultaneity of flow are there- 
fore distinct but complementary things. Without simultaneity 
of flow, we would not consider these three terms interchange- 
able: continuity of our inner life, continuity of a voluntary 
motion which our mind indefinitely prolongs, and continu- 
ity of any motion through space. Real duration and spatia- 
lized time would not then be equivalent, and consequently 
time in general would no longer exist for us; there would 
be only each one's duration. But, on the other hand, this 
time can be computed thanks only to the simultaneity of 
the instant. We need this simultaneity of the instant in order 
(1) to note the simultaneity of a phenomenon with a clock 
moment, (2) to point off, all along our own duration, the 
simultaneities of these moments with moments of our dura- 
tion which are created in the very act of pointing. Of these 
two acts, the first is the essential one in the measurement of 
time. But without the second, we would have no particular 
measurement, we would end up with a figure t representing 
anything at all, we would not be thinking of time. It is there- 
fore the simultaneity between two instants of two motions 
outside of us that enables us to measure time; but it is the 
simultaneity of these moments with moments pricked by them 
along our inner duration that makes this measurement one 
of time. 

We shall have to dwell upon these two points. But let us 
first open a parenthesis. We have just distinguished between 
two "simultaneities of the instant"; neither of the two is the 
simultaneity most in question in the theory of relativity, 
namely, the simultaneity between readings given by two sep- 
arated clocks. Of that we have spoken in our first chapter; 
we shall soon be especially occupied with it. But it is clear 
that the theory of relativity itself cannot help acknowledging 
the two simultaneities that we have just described; it confines 



itself to adding a third, one that depends upon a synchroniz- 
ing of clocks. Now we shall no doubt show how the readings 
of two separated clocks C and C, synchronized and showing 
the same time, are or are not simultaneous according to one's 
point of view. The theory of relativity is correct in so stating; 
we shall see upon what condition. But it thereby recognizes 
that an event E occurring beside clock C is given in simul- 
taneity with a reading on clock C in a quite different sense- 
in the psychologist's sense of the word simultaneity. And like- 
wise for the simultaneity of event £' with the reading on its 
"neighboring" clock C. For if we did not begin by admitting 
a simultaneity of this kind, one which is absolute and has 
nothing to do with the synchronizing of clocks, the clocks 
would serve no purpose. They would be bits of machinery 
with which we would amuse ourselves by comparing them with 
one another; they would not be employed in classifying events; 
in short, they would exist for their own sake and not to serve 
us. They would lose their raison d'etre for the theoretician 
of relativity as for everyone else, for he too calls them in only 
to designate the time of an event. Now, it is very true that 
simultaneity thus understood is easily established between 
moments in two flows only if the flows pass by "at the same 
place." It is also very true that common sense and science 
itself until now have, a priori, extended this conception of 
simultaneity to events separated by any distance. They no 
doubt imagined, as we said further back, a consciousness coex- 
tensive with the universe, capable of embracing the two events 
m a unique and instantaneous perception. But, more than 
anything else, they applied a principle inherent in every 
mathematical representation of things and asserting itself in 
the theory of relativity as well. We find in it the idea that the 
distinction between "small" and "large," "not far apart" and 
"very far apart," has no scientific validity and that if we can 
speak of simultaneity outside of any synchronizing of clocks, 
independently of any point of view, when dealing with an 
event and a clock not much distant from one another, we have 
this same right when the distance is great between the clock 



and the event or between the two clocks. No physics, no 
astronomy, no science is possible if we deny the scientist the 
right to represent the whole universe schematically on a piece 
of paper. We therefore implicitly grant the possibility of reduc- 
ing without distorting. We believe that size is not an absolute, 
that there are only relations among sizes, and that everything 
would turn out the same in a universe made smaller at will, 
if the relations among parts were preserved. But in that case 
how can we prevent our imagination, and even our under- 
standing, from treating the simultaneity of the readings of two 
very widely separated clocks like the simultaneity of two clocks 
slightly separated, that is, situated "at the same place"? A 
thinking microbe would find an enormous interval between 
two "neighboring" clocks. And it would not concede the exist- 
ence of an absolute, intuitively perceived simultaneity between 
their readings. More Einsteinian than Einstein, it would see 
simultaneity here only if it had been able to note identical 
readings on two microbial clocks, synchronized by optical sig- 
nals, which it had substituted for our two "neighboring" 
clocks. Our absolute simultaneity would be its relative simul- 
taneity because it would refer our absolute simultaneity to the 
readings on its two microbial clocks which it would, in its 
turn, perceive (which it would, moreover, be equally wrong to 
perceive) "at the same place." But this is of small concern at 
the moment; we are not criticizing Einstein's conception; we 
merely wish to show to what we owe the natural extension 
that has always been made of the idea of simultaneity, after 
having actually derived it from the ascertainment of two 
neighboring" events. This analysis, which has until now 
hardly been attempted, reveals a fact that the theory of rela- 
tivity could make use of. We see that if our understanding 
passes here so easily from a short to a long distance, from 
simultaneity between neighboring events to simultaneity be- 
tween widely-separated events, if it extends to the second case 
he absolute character of the first, it is because it is accustomed 
to believing that we can arbitrarily modify the dimensions of 
all things on condition of retaining their relations. But it is 


time to close the parenthesis. Let us return to the intuitively 
perceived simultaneity which we first mentioned and the two 
propositions we had set forth: (1) it is the simultaneity be- 
tween two instants of two motions outside us that allows us 
to measure an interval of time; (2) it is the simultaneity of 
these moments with moments dotted by them along our inner 
duration that makes this measurement one of time [pp. 52-54]. 

The first point is obvious. We saw above how inner duration 
exteriorizes itself as spatialized time and how the latter, space 
rather than time, is measurable. It is henceforth through the 
intermediary of space that we shall measure every interval of 
time. As we shall have divided it into parts corresponding to 
equal spaces, equal by definition, we shall have at each divi- 
sion point an extremity of the interval, an instant, and we 
shall regard the interval itself as the unit of time. We shall 
then be able to consider any motion, any change, occurring 
beside this model motion; we shall point off the whole length 
of its unfolding with "simultaneities of the instant." As many 
simultaneities as we shall have established, so many units of 
time shall we record for the duration of the phenomenon. 
Measuring time consists therefore in counting simultaneities. 
All other measuring implies the possibility of directly or indi- 
rectly laying the unit of measurement over the object meas- 
ured. All other measuring therefore bears upon the interval 
between the extremities even though we are, in fact, confined 
to counting these extremities. But in dealing with time, we can 
only count extremities; we merely agree to say that we have 
measured the interval in this way. If we now observe that 
science works exclusively with measurements, we become aware 
that, with respect to time, science counts instants, takes note of 
simultaneities, but remains without a grip on what happens 
m the intervals. It may indefinitely increase the number of 
extremities, indefinitely narrow the intervals; but always the 
interval escapes it, shows it only its extremities. If every motion 
ln the universe were suddenly to accelerate in proportion, in- 
cluding the one that serves as the measure of time, something 
w ould change for a consciousness not bound up with intra- 



cerebral molecular motions; it would not receive the same 
enrichment between sunup and sundown; it would therefore 
detect a change; in fact, the hypothesis of a simultaneous ac- 
celeration of every motion in the universe makes sense only if 
we imagine a spectator-consciousness whose completely quali- 
tative duration admits of a more or a less without being 
thereby accessible to measurement. 6 But the change would 
exist only for that consciousness able to compare the flow of 
things with that of the inner life. In the view of science noth- 
ing would have changed. Let us go further. The speed of un- 
folding of this external, mathematical time might become 
infinite; all the past, present, and future states of the universe 
might be found experienced at a stroke; in place of the un- 
folding there might be only the unfolded. The motion repre- 
sentative of time would then have become a line; to each of 
the divisions of this line there would correspond the same 
portion of the unfolded universe that corresponded to it before 
in the unfolding universe; nothing would have changed in the 
eyes of science. Its formulae and calculations would remain 
what they were. 

It is true that exactly at the moment of our passing from the 
unfolding to the unfolded, it would have been necessary to 
endow space with an extra dimension. More than thirty years 

« It is obvious that our hypothesis would lose its meaning if we thought 
ot consciousness as an "epiphenomenon" added to cerebral phenomena of it would be merely the result or expression. We cannot dwell here 
upon this theory of consciousness-as-epiphenomenon, which we tend more 
and more to consider arbitrary. We have discussed it in detail in several 
LT W ° r ^' n ° tably in thE first three ^apters of Matiire el mSmoire 
ZZ7/ , Ty) 3nd in difFerent e ^ys ^ L'Energie spiritud* 
(Mmd-Energy). Let us confine ourselves to recalling: (1) that this theory 

made 0 out y mT ^ ^ (2) that its metaphysical origins are easily 
J Zfl • takCn 1Uerall y> " would be self-contradictory. (Concern- 
ween' P ° mt and the °""l"ion, which the theory implies be- 
Part F ° C0ntr 7 asserti °™. see L'Energie spirituelle (Mind-Energ?) 
ciou nel; a Akan ' 1919) ' PP- 203 - 2 23- In the present work, we take con- 
and ori^ "V™"* ^es it to us, without theorizing about its nature 



ago, 7 we pointed out that spatialized time is really a fourth 
dimension of space. Only this fourth dimension allows us to 
juxtapose what is given as succession: without it, we would 
have no room. Whether a universe has three, two, or a single 
dimension, or even none at all and reduces to a point, we can 
always convert the indefinite succession of all its events into 
instantaneous or eternal juxtaposition by the sole act of grant- 
ing it an additional dimension. If it has none, reducing to a 
point that changes quality indefinitely, we can imagine the 
rapidity of succession of the qualities becoming infinite and 
these points of quality being given all at once, provided we 
bring to this world without dimension a line upon which the 
points are juxtaposed. If it already had one dimension, if it 
were linear, two dimensions would be needed to juxtapose the 
lines of quality— each one indefinite— which were the succes- 
sive moments of its history. The same observation again if it 
had two dimensions, if it were a surface universe, an indefinite 
canvas upon which flat images would indefinitely be drawn, 
each one covering it completely; the rapidity of succession of 
these images will again be able to become infinite, and we 
shall again go over from a universe that unfolds to an un- 
folded universe, provided that we have been accorded an extra 
dimension. We shall then have all the endless, piled-up can- 
vasses giving us all the successive images that make up the 
entire history of the universe; we shall possess them all to- 
gether; but we shall have had to pass from a flat to a volumed 
universe. It is easy to understand, therefore, why the sole act 
of attributing an infinite speed to time, of substituting the 
unfolded for the unfolding, would require us to endow our 
solid universe with a fourth dimension. Now, for the very 
reason that science cannot specify the "speed of unfolding" of 
tune, that it counts simultaneities but necessarily neglects 
intervals, it deals with a time whose speed of unfolding we 

' Essa * sur les donnies immidiates de la conscience (Time and Free 
W M), p. 83. 


may as well assume to be infinite, thereby virtually conferring 
an additional dimension upon space. 

Immanent in our measurement of time, therefore, is the 
tendency to empty its content into a space of four dimensions 
in which past, present, and future are juxtaposed or superim- 
posed for all eternity. This tendency simply expresses our in- 
ability mathematically to translate time itself, our need to 
replace it, in order to measure it, by simultaneities which we 
count. These simultaneities are instantaneities; they do not 
partake of the nature of real time; they do not endure. They 
are purely mental views that stake out conscious duration and 
real motion with virtual stops, using for this purpose the 
mathematical point that has been carried over from space to 

But if our science thus attains only to space, it is easy to see 
why the dimension of space that has come to replace time is 
still called time. It is because our consciousness is there. It 
infuses living duration into a time dried up as space. Our 
mind, interpreting mathematical time, retraces the path it has 
traveled in obtaining it. From inner duration it had passed to 
a certain undivided motion which was still closely bound up 
with it and which had become the model motion, the genera- 
tor or computer of time; from what there is of pure mobility 
in this motion, that mobility which is the link between motion 
and duration, it passed to the trajectory of the motion, which 
is pure space; dividing the trajectory into equal parts, it passed 
from the points of division of this trajectory to the correspond- 
ing or "simultaneous" points of division of the trajectory of 
any other motion. The duration of this last motion was thus 
measured; we have a definite number of simultaneities; this 
will be the measure of time; it will henceforth be time itself. 
But this is time only because we can look back at what we 
have done. From the simultaneities staking out the continuity 
of motions, we are always prepared to reascend the motions 
themselves and, through them, the inner duration that is con- 
temporaneous with them, thus replacing a series of simultanei- 
ties of the instant, which we count but which are no longer 



time, by the simultaneity of flows that leads us back to inner, 
real duration. 

Some will wonder whether it is useful to return to it, and 
whether science has not, as a matter of fact, corrected a mental 
imperfection, brushed aside a limitation of our nature, by 
spreading out "pure duration" in space. These will say: "Time, 
which is pure duration, is always in the course of flowing; we 
apprehend only its past and its present, which is already past; 
the future appears closed to our knowledge, precisely because 
we believe it open to our action— it is the promise or anticipa- 
tion of unforeseeable novelty. But the operation by which we 
convert time into space for the purpose of measuring it in- 
forms us implicitly of its content. The measurement of a thing 
is sometimes the revealer of its nature, and precisely at this 
point mathematical expression turns out to have a magical 
property: created by us or risen at our bidding, it does more 
than we asked of it; for we cannot convert into space the time 
already elapsed without treating all of time the same way. The 
act by which we usher the past and present into space spreads 
out the future there without consulting us. To be sure, this 
future remains concealed from us by a screen; but now we 
have it there, all complete, given along with the rest. Indeed, 
what we called the passing of time was only the steady sliding 
of the screen and the gradually obtained vision of what lay 
waiting, globally, in eternity. Let us then take this duration 
for what it is, for a negation, a barrier to seeing all, steadily 
pushed back; our acts themselves will no longer seem like a 
contribution of unforeseeable novelty. They will be part of 
the universal weave of things, given at one stroke. We do not 
introduce them into the world; it is the world that introduces 
them ready-made into us, into our consciousness, as we reach 
them. Yes, it is we who are passing when we say time passes; 
it is the motion before our eyes which, moment by moment, 
actualizes a complete history given virtually." Such is the meta- 
physic immanent in the spatial representation of time. It is 
inevitable. Clear or confused, it was always the natural meta- 
Physic of the mind speculating upon becoming. We need not 



discuss it here, still less replace it by another. We have ex- 
plained elsewhere why we see in duration the very stuff of our 
existence and of all things, and why, in our eyes, the universe 
is a continuity of creation. We thus kept as close as possible 
to the immediate; we asserted nothing that science could not 
accept and use; only recently, in an admirable book, a philoso- 
pher-mathematician affirmed the need to admit of an "advance 
of Nature" and linked this conception with ours. 8 For the 
present, we are confining ourselves to drawing a demarcation 
line between what is theory, metaphysical construction, and 
what is purely and simply given in experience; for we wish to 
keep to experience. Real duration is experienced; we learn 
that time unfolds and, moreover, we are unable to measure it 
without converting it into space and without assuming all we 
know of it to be unfolded. But, it is impossible mentally to 
spatialize only a part; the act, once begun, by which we unfold 
the past and thus abolish real succession involves us in a total 
unfolding of time; inevitably we are then led to blame human 
imperfection for our ignorance of a future that is present and 
to consider duration a pure negation, a "deprivation of eter- 
nity. ' Inevitably we come back to the Platonic theory. But 
since this conception must arise because we have no way of 
limiting our spatial representation of elapsed time to the past, 
it is possible that the conception is erroneous, and in any case 
certain that it is purely a mental construction. Let us there- 
tore keep to experience. 

^-iSS p« 0 ; ? h e Conc r, : f Nature ( c - brid * e: f c T 

tivitv in t ~ , f )- Thls work ( which ta kes the theory of rela- 

t^l^: :z:^ h T e r the most ever Jiia z z 

Whitph^H-; , ! ^ 6 Televant Passage occurs on page 54 of 
oTn tu thaT I , ^ 38 f ° ll0WS: an exhibition of the process 

1 a l be tP f mtiaa ha PP ens *»d passes. The process of nature 
from " in ! h ?! ' PaSSage ° £ natUre/ 1 defini *ly -Sain at this stage 
SvLd Ufe t ^ SinCC the arable time of science and of 

mental \^JT Y ^ eXhibUs some as P<** °* *e ™ re tm6A ' 
Tn M accord ! £T*° ° f natUrC - 1 believe tha < this doctrine I a* 
L^T?«7* B ? W th0U 6 h he »- 'ti'ne' for the fundamental 
which I call the 'passage of nature." "] 



If time has a positive reality, if the delay of duration at in- 
stantaneity represents a certain hesitation or indetermination 
inherent in a certain part of things which holds all the rest 
suspended within it; in short, if there is creative evolution, 
I can very well understand how the portion of time already 
unfolded may appear as juxtaposition in space and no longer 
as pure succession; I can also conceive how every part of the 
universe which is mathematically linked to the present and 
past— that is, the future unfolding of the inorganic world- 
may be representable in the same schema (we once demon- 
strated that in astronomical and physical matters prevision is 
really a vision). We believe that a philosophy in which dura- 
tion is considered real and even active can quite readily admit 
Minkowski's and Einstein's space-time (in which, it must be 
added, the fourth dimension called time is no longer, as in our 
examples above, a dimension completely similar to the others). 
On the other hand, you will never derive the idea of a tem- 
poral flow from Minkowski's schema. Is it not better, in that 
case, to confine ourselves, until further notice, to that one of 
the two points of view which sacrifices nothing of experi- 
ence, and therefore— not to prejudge the question— nothing 
of appearances? Besides, how can a physicist wholly reject 
inner experience if he operates with perceptions and, there- 
fore, with the data of consciousness? It is true that a certain 
doctrine accepts the testimony of the senses, that is, of con- 
sciousness, in order to obtain terms among which to establish 
relations, then retains only the relations and regards the terms 
as nonexistent. But this is a metaphysic grafted upon science, 
it is not science. And, to tell the truth, it is by abstraction that 
we distinguish both terms and relations: a continual flow from 
which we simultaneously derive both terms and relations and 
which is, over and above all that, fluidity; this is the only 
immediate datum of experience. 

But we must close this overly long parenthesis. We believe 
we have achieved our purpose, which was to describe the 
salient features of a time in which there really is succession. 
Abolish these features and there is no longer succession, but 


juxtaposition. You can say that you are still dealing with 
time— we are free to give words any meaning we like, as long 
as we begin by defining that meaning— but we shall know that 
we are no longer dealing with an experienced time; we shall 
be before a symbolic and conventional time, an auxiliary 
magnitude introduced with a view to calculating real magni- 
tudes. It is perhaps for not having first analyzed our mental 
view of the time that flows, our feeling of real duration, that 
there has been so much trouble in determining the philosoph- 
ical meaning of Einstein's theories, that is, their relation to 
reality. Those whom the paradoxical appearance of the the- 
ories inconvenienced have declared Einstein's multiple times 
to be purely mathematical entities. But those who would like 
to dissolve things into relations, who regard every reality, even 
ours, as a confusedly perceived mathematics, are apt to declare 
that Minkowski's and Einstein's space-time is reality itself, 
that all of Einstein's times are equally real, as much and per- 
haps more so than the time that flows along with us. We are 
too hasty in both instances. We have just stated, and we shall 
soon demonstrate in greater detail, why the theory of rela- 
tivity cannot express all of reality. But it is impossible for it 
not to express some. For the time that intervenes in the 
Michelson-Morley experiment is a real time-real again is the 
time to which we return with the application of the Lorentz 
tormulae. If we leave real time to end with real time, we have 
perhaps made use of mathematical artifices in between, but 
uiese must have some connection with things. It is therefore 
a question of allotting shares to the real and to the conven- 

forAis usk 311317568 WCre Simply intended to P ave the Way 

folW^ 6 h r,, jUSt U " ered the word "reality"; and in what 
eaf W < ^ ^nstantly be speaking of what is real and not 
defined ,V by that? If * ^re necessary to 

wl cLld y / n , general> l ° sa ? ^ wh *t «gn we recognize it, 

SoT^-r u S ° With ° Ut dassif ™ Selves within * 
school, philosophers are not in agreement, and the problem 



has received as many solutions as there are shades of realism 
and idealism. We would, besides, have to distinguish between 
the standpoints of philosophy and science; the former rather 
regards the concrete, all charged with quality, as the real; the 
latter extracts or abstracts a certain aspect of things and retains 
only size or relation among sizes. Very happily, we have only 
to be occupied, in all that follows, with a single reality, time. 
This being so, it will be easy for us to follow the rule we have 
imposed upon ourselves in the present essay, that of advancing 
nothing that cannot be accepted by any philosopher or sci- 
entist—even nothing that is not implied in all philosophy 
and science. 

Everyone will surely agree that time is not conceived with- 
out a before and an after— time is succession. Now we have just 
shown that where there is not some memory, some conscious- 
ness, real or virtual, established or imagined, actually present 
or ideally introduced, there cannot be a before and an after; 
there is one or the other, not both; and both are needed to 
constitute time. Hence, in what follows, whenever we shall 
wish to know whether we are dealing with a real or an 
imaginary time, we shall merely have to ask ourselves whether 
the object before us can or cannot be perceived, whether we 
can or cannot become conscious of it. The case is privileged; it 
is even unique. If it is a question of color, for example, con- 
sciousness undoubtedly intervenes at the beginning of the 
study in order to give the physicist the perception of the thing; 
but the physicist has the right and the duty to substitute for 
the datum of consciousness something measurable and numer- 
able with which he will henceforward work while granting 
" the name of the original perception merely for greater con- 
venience. He can do so because, with this original perception 
eliminated, something remains, or at the very least, is deemed 
to remain. But what will be left of time if you take succession 
out of it? And what is left of succession if you remove even 
the possibility of perceiving a before and an after? I grant you 
the right to substitute, say, a line for time, since to measure it 
is quite in order. But a line can be called time only when the 


juxtaposition it affords is convertible into 
wise you are arbitrarily and conventionally giving that line 
Z name of time. We must be forewarned of thi. so as no 
to lay ourselves open to a serious error. What will happen 
you introduce into your reasoning and figuring the hypothes s 
that the thing you called "time" cannot, on pain of contra- 
diction, be perceived by a consciousness, either real or imagi- 
nary? Will you not then be working, by definition, with an 
imaginary, unreal time? Now such is the case with the times 
with which we shall often be dealing in the theory of rela- 
tivity. We shall meet with perceived or perceptible ones-those 
will be considered real. But there are others that the theory 
prohibits, as it were, from being perceived or becoming per- 
ceptible; if they became so, they would change in scale, so that 
measurement, correct if it bears upon what we do not perceive, 
would be false as soon as we do perceive. Why not declare 
these latter unreal, at least as far as their being "temporal 
goes? I admit that the physicist still finds it convenient to call 
them time; we shall soon see why. But if we liken these times 
to the other, we fall into paradoxes that have certainly hurt 
the theory of relativity, even if they have helped popularize it. 
It will therefore be no surprise if, in the present study, we 
require the property of being perceived or perceptible for 
everything held up as real. We shall not be deciding the ques- 
tion of whether all reality possesses this salient feature. We are 
only dealing here with the reality of time. 


Concerning the Plurality of Times 

The multiple, slowed times of the theory of relativity: 
why they are compatible with a single, umversal time; 
"learned" simultaneity, dislocatable into successum: why 
it is compatible with the natural, "intuUxve simultane- 
ity; examination of the paradoxes of time; the hypo** 
sil of the passenger in a projectile; Minkowski s schema, 
the confusion that is the source of all the paradoxes 

Let us then finally turn to Einstein's time, g oin ^ a f ^ 
everything we said when at first we assumed a ™; 
Here is the earth in motion in its orbit, and, on it, the Mitfu* 
son-Morley apparatus. The experiment is Vf 0 ™™' " 
begun again at different times of the year ^ conseo^ently 
for different speeds of our planet. Always the bea« of^gh 
behaves as if the earth were motionless. Such is the fact. What 

18 ButTrlrwhT'speak of speeds of our planet? Is the* Mhe 

earth, absolutely speaking, in motion through spac ? Of cour e 

not; we are at the standpoint of relativity and 

absolute motion. When you speak of the orbit des cribed br * e 

earth, you are placing yourself at an arbitrarily chosen point 

of view", that rf the inhabitants of the sun (o a ™J>™™ 

habitable). It suits you to adopt this system of re ere ^ ^ 

*hy should the beam of light shot against he ^mrrr 

apparatus take your whim into account? I ^ all tn ^ 

occurs is the reciprocal displacement of the eartn a 

we can take the L, the earth, or any other observation post 


as our system of reference. Let us choose the earth. The prob- 
lem disappears with regard to it. We need no longer wonder 
why the interference bands preserve the same appearance, why 
the same result is observed at any time of the year. Quite sim- 
ply, it is because the earth is motionless. 

It is true that, in our eyes, the problem then reappears with 
regard to the inhabitants of the sun. I say "in our eyes," be- 
cause, to a solar physicist, the question will no longer concern 
the sun; it is now the earth that is moving. In short, each of 
the two physicists will still pose the problem for the system 
that is not his. 

Each of them will find himself with respect to the other in 
the situation Peter was in earlier with regard to Paul. Peter 
was stationed in the motionless ether; he lived in a privileged 
system S. He saw Paul, borne along in the motion of the mov- 
ing system S', performing the same experiment as he did and 
obtaining the same speed for light, even though this speed 
ought to have been reduced by that of the moving system. The 
matter was explained by the slowing of time, the contractions 
in length and the breakup of simultaneity that motion brought 
about in system S'. Now, no more absolute motion and there- 
fore no more absolute rest: each of the two systems in re- 
ciprocal displacement is immobilized in turn by the ruling 
that gives it the status of a system of reference. But, all the 
while this convention is maintained, we shall be able to repeat 
about the immobilized system what was said before about the 
actually stationary system, and about the mobilized system, 
what applied to the moving system actually traveling through 
the ether. In order to fix our ideas, let us again give the titles 
of S and S' to two systems in mutual displacement. And, to 
simplify things, let us assume that the whole universe reduces 
to these two systems. If S is the system of reference, the physi- 
cist located ln S, bearing in mind that his colleague in S' finds 
the same speed for light as he, interprets the result as we did 
above. He renects: "The system travels at speed v with respect 
to me, motionless. But, the Michelson-Morley experiment gives 
the same result over there as here. The truth is, therefore, that, 


as a result of motion, a contraction takes place in the direction 
of the system's motion: a length / becomes I yjl-^- More- 
over, an expansion of time is linked to this contraction of 
lengths; where a clock in S' ticks off a f number of seconds, 


there has really elapsed of them. Finally, when the 

clocks in S', placed at intervals along its direction of motion 
and separated by distances of I, point to the same time, I see 
that the signals going and coming between two consecutive 
clocks do not make the same trip on leaving as on returning, 
as a physicist inside system S' and unaware of its motion be- 
lieves; when these clocks show him a simultaneity, they are 

really pointing to successive moments separated by — of his . i 

c " ' 1 

clock's seconds and, therefore by - seconds of mine." , » 

Such would be the reasoning of the physicist in S. And, build- jj; 
ln g up a unified mathematical representation of the universe, f p 

he would make use of the space and time measurements of 
his colleague in system 5' only after having made them $ 
undergo the Lorentz transformation. "jZ; 

But the physicist in system S' would proceed in exactly the ' j: 

same way. Ruling himself motionless, he would repeat of S " t 

everything that his colleague located in S would have said '(] 
about S'. In the mathematical representation of the universe )- 
which he would build up, he would consider the measure- 
ments that he himself would have taken within his own system 
as being exact and definitive but would correct in accordance 
w ith the Lorentz formulae all those which would have been 
taken by the physicist attached to system S. 

Thus, two mathematical representations of the universe 
Would be obtained, completely different from one another if 
We consider the figures appearing in them, identical if we take 
mto account the relations among phenomena which they indi- 



cate-relations that we call the laws of nature. That difference 
is, moreover, the very condition of this identity. When we take 
different photographs of an object while walking around it, 
the variability of the details only expresses the invariability of 
their interrelations, in other words, the permanence of the 

Here we are, then, brought round again to multiple times, 
to simultaneities that are successions, and to successions that 
are simultaneities, to lengths that must be measured differendy 
according to whether they are ruled stationary or moving. But 
this time we are before the definitive form of the theory of 
relativity. We must ask ourselves how these words are to be 

Let us first consider the plurality of times, going back to our 
two systems S and S'. The physicist situated in S adopts his 
system as the system of reference. There they are, then, S at 
rest and S' in motion. Inside this system ruled motionless, our 
physicist begins the Michelson-Morley experiment. To attain 
our presently limited aim it will be useful to cut the experi- 
ment in two and to hold on to only half of it, if we may so 
express ourselves. We shall therefore assume that the physicist 
is occupied only with the journey of light in the direction OB 
perpendicular to that of the reciprocal motion of the two sys- 
tems. On a clock located at point O, he reads the time t that 
the beam has taken to go from O to B and back again. What 
kind of time are we dealing with? 

With a real time, of course, in the meaning that we gave 
above to this expression. Between the beam's departure and 
return the physicist's consciousness has lived a certain dura- 
tion; the motion of the clock hands is a flow contemporaneous 
with this inner flow and serves to measure it. On this point 
there is no doubt or difficulty. A time lived and recorded by 
a consciousness is real by definition 

hin^u S C i nSidCr a XCOnd P h y sicis * Seated in S'. He rules 
system TT ' bdng USCd l ° hi * °™ *y stem aS 

Moriev 1^ ThCre he is ' Pertoming the Michelson- 

Morley experiment or, rather, he too, only half of it. On a 


clock placed at O', he notes the time that the beam of light 
takes to go from O' to B' and back again. What, then, is this 
time that he records? The time that he lives, of course. The 
motion of his clock is contemporaneous with the flow of his 
consciousness. It is, again, a real time by definition. 

Thus, the time lived and recorded by the first physicist in 
his system and the time lived and recorded by the second one 
in his are both real times. 

Are they both one and the same time? Are they different 
times? We are going to demonstrate that we are dealing with 
the same time in both cases. 

Indeed, whatever the meaning we assign to the slowings or 
accelerations of time, and therefore to the multiple times that 
are in question in the theory of relativity, one thing is certain: 
these slowings and accelerations are due only to the motions 
of the systems we are considering and are subject only to the 
speed with which we imagine each system propelled. We are 
therefore changing nothing in any time whatever, real or 
imaginary, in system S', if we assume that this system is a 
duplicate of system S; for the system's content, the nature of 
the events that unfold in it, are extraneous; only the system's 
speed of translation matters. But if S' is a double of S, it is 
obvious that the time lived and noted by the second physicist 
during his experiment in system S', judged motionless by him, 
is identical with the time lived and noted by the first in system 
S likewise judged motionless, since 5 and S', once immobilized, 
are interchangeable. Hence, the time lived and recorded in the 
system, the time inside of and immanent in the system, in 
short, real time, is the same for S and S'. 

But what then are the multiple times with their unequal 
speeds of flow which the theory of relativity finds in different 
systems in accordance with the speed with which these systems 
are propelled? 

Let us return to our two systems 5 and S'. If we consider the 
time which the physicist Peter, situated in S, attributes to sys- 
tem S', we see that this time is, indeed, slower than the time 
recorded by Peter in his own system. The former time is there- 


fore not lived by Peter. But we know that it is not lived by 
Paul either. It is therefore not lived either by Peter or Paul. 
With even more reason is it not lived by others. But this is 
not saying enough. If the time attributed by Peter to Paul s 
system is not lived by Peter, Paul, or anyone, is it at least 
conceived by Peter as lived or able to be lived by Paul, or, 
more generally, by someone, or still more generally by some- 
thing? Looking closely, we see that it is nothing of the kind. 
To be sure, Peter pastes a label on this time with Paul's name 
on it; but if he were picturing a conscious Paul, living his own 
duration and measuring it, he would by that very act see Paul 
take his own system as system of reference and therefore 
take his place within this single time, inside each system, to 
which we have just referred; by that very act, moreover, Peter 
would also take temporary leave of his system of reference, 
consequently, of his existence as a physicist, and consequently, 
of his consciousness as well; Peter would no longer see himself 
as anything but a vision of Paul's. But when Peter attributes 
a slowed time to Paul's system, he is no longer thinking of 
Paul as a physicist, nor even a conscious being. He is emptying 
Paul's visual image of its inner, living consciousness, retaining 
of the person only its outer envelope (it alone, in fact, is of 
interest to physics). Then, Peter takes the figures by which 
Paul would have designated the time intervals of his own sys- 
tem, were he conscious, and multiplies them by — — === s0 as 

to make these figures fit into a mathematical representation of 
the universe conceived from his own point of view and no 
longer from Paul's. Thus, to sum up, whereas the time at- 
tributed by Peter to his own system is a time he has lived, the 
time he attributes to Paul's is neither a time that either Peter 
or Paul has lived, nor a time that Peter conceives as lived or as 
capable of being lived by a living, conscious Paul. What is it> 
then, if not a mere mathematical expression meant to indicate 
that Peter's not Paul's system has been taken as the system 
of reference? 


I am an artist and I have to portray two subjects, John and 
James, the one standing next to me and the other, two or 
three hundred yards away. I draw the former life-size and 
shrink the latter to the size of a midget. A fellow artist stand- 
ing next to James and also desirous of painting the two will 
proceed inversely; he will show John very small and James in 
normal size. We shall, moreover, both be right. But because we 
are both right, are we therefore justified in concluding that 
John and James have neither normal nor a midget's stature, 
or that they have both at once, or anything we like? Of course 
not. Shape and size are terms that have an exact meaning in 
connection with a posed model; it is what we perceive of the 
height and width of an individual when we are standing next 
to him, when we can touch him and measure his body with 
a ruler. Being next to John, measuring him if I like and in- 
tending to paint him in his normal height, I grant him his real 
size; and, in portraying James as a midget, I am simply express- 
ing the impossibility of my touching him-even, if we may be 
permitted to say so, the degree of this impossibility; the degree 
of impossibility is exactly what is called distance, and it is 
distance for which perspective makes allowance. In the same 
way, in the system in which I live and which I mentally im- 
mobilize by conceiving as a system of reference, I directly 
measure a time that is mine and my system's; it is this meas- 
urement which I inscribe in my mathematical representation 
of the universe for all that concerns my system. But in im- 
mobilizing my system, I have set the others moving, and I have 
set them moving variously. They have acquired different 
speeds. The greater their speed, the further removed they are 
from my immobility. It is this greater or lesser distance of 
their speed from my zero speed which I express in my mathe- 
matical representation of other systems when I assign them 
more or less slowed times, all, of course, slower than mine, 
just as it is the greater or lesser distance between James and 
me which I express by shrinking his figure more or less. The 
multiplicity of times which I thus obtain does not preclude 
the unity of real time; rather, it presupposes it, in the same 



way that the diminution of James's figure with distance, on 
a series of canvases in which I would show him more or less 
distant, indicates that James remains the same size. 

Thus is effaced the paradoxical form given the theory of 
the plurality of times. "Imagine," we are told, "a passenger 
in a projectile launched from the earth at about one twenty- 
thousandth less than the speed of light, which meets a star and 
returns to the earth at the same speed. Having aged, say, two 
years up to the time he gets out of his vehicle, he discovers 
that our globe has aged two hundred years." Are we really 
sure of this? Let us look more closely. We shall see the mirage 
effect vanish, for it is nothing else. 

The projectile has been fired from a cannon attached to the 
motionless earth. Let Peter be the one who remains beside the 
cannon, the earth then becoming our system S. The passenger 
in the projectile S' then becomes Paul. The theory has been 
advanced, we said, that Paul would return after two hundred 
years lived by Peter. Peter has therefore been considered living 
and conscious; two hundred years of his inner flow have really 
elapsed for Peter between the departure and return of Paul. 

Let us now turn to Paul. We wish to know how much time 
he has lived. It is therefore to the living, conscious Paul that 
we must address ourselves and not to Paul's image represented 
in Peter's consciousness. But the living, conscious Paul ob- 
viously takes his vehicle as his system of reference; in that 
very act, he immobilizes it. As soon as we address ourselves to 
Paul, we are with him, we adopt his point of view. But then, 
presto, the projectile has stopped; it is the cannon, with the 
earth attached, which flies through space. We must now repeat 
for Paul everything we said about Peter: since motion is re- 
ciprocal, the two people are interchangeable. If, earlier, look- 
ing into Peter's consciousness, we witnessed a certain flow, we 
are going to find exactly the same flow in Paul's consciousness. 
If we said that the first flow lasted two hundred years, the 
other flow will also last two hundred years. Peter and Paul, 
earth and projectile, will have gone through the same dura- 
tion and aged equally. 



Where then are the two years of slowed time which were 
gently to idle by for the projectile while two hunderd years 
would have to race past on the earth? Has our analysis vapor- 
ized them? Not at all! We are going to rediscover them. But 
we shall no longer be able to lodge anything in them, neither 
beings nor things; and we shall have to look for another way 
not to grow old. 

Our two people have actually seemed to be living two hun- 
dred years at one and the same time because we placed our- 
selves at both their viewpoints. This was necessary in order to 
interpret philosophically Einstein's thesis, which is that of the 
radical relativity and, therefore, the perfect reciprocity of recti- 
linear, uniform motion. 1 But this procedure is proper to the 
philosopher who takes Einstein's thesis in its wholeness and 
attaches himself to the reality— I mean the perceived or per- 
ceptible thing— which this thesis plainly expresses. It involves 
not for a moment losing sight of the idea of reciprocity and, 
consequently, going unceasingly from Peter to Paul and from 
Paul to Peter, considering them interchangeable, immobilizing 
them by turns, immobilizing them, moreover, for only an 
instant, thanks to a rapid oscillation of the attention that does 
not wish to give up anything of the thesis of relativity. But 
the physicist is obliged to proceed otherwise, even if he adheres 
unreservedly to Einstein's theory. He unquestionably begins by 
aligning himself with it. He affirms reciprocity. He grants that 
we have the choice between Peter's and Paul's point of view. 
But, having granted this, he chooses one of the two, for he 
cannot refer events in the universe simultaneouly to two sys- 
tems with different axes. If he puts himself mentally in Peter's 
Place, he will record for Peter the time that Peter records for 
himself, namely, the time really lived by Peter, and for Paul 
the time that Peter attributes to him. If he is with Paul, he 
will record for Paul the time that Paul records for himself, 

^he motion of the projectile can be considered rectilinear and : uni- 
form during both its outbound and inbound journeys. This is all that is 
Quired for the validity of the argument just advanced. See Appendix 
at the end of this volume, p. 163. 



namely, the time that Paul actually lives, and for Peter, the 
time that Paul confers upon him. But, once again, he will of 
necessity decide between Peter and Paul. Suppose he chooses 
Peter. It is in that case really two years, and only two years, 
that he must record for Paul. 

The fact is that Peter and Paul are involved with the same 
physics. They observe the same relations among phenomena, 
discover the same laws in nature. But Peter's system is motion- 
less and Paul's is in motion. As long as we are dealing with 
phenomena in some way attached to the system, that is, so 
defined by physics that the system is deemed to be carrying 
them along when it is ruled in motion, the laws of these 
phenomena must plainly be the same for both Peter and Paul: 
phenomena m motion, being perceived by Paul who is en- 
dowed with the same motion as they, are motionless for him 

sv^T ^ T Cdy aS analo g°™ phenomena in Peter's 
system do to Peter. But electromagnetic phenomena arise in 

T CVen th ° Ugh ^ s y stem in which they occur 
it* rnoT T^ g ' W Can n ° lon S er consider as sharing 

. 1 .° n ; And yet the interrelations of these phenomena, 

tern's rf, thC P henome ^ carried along in the sys- 

ZLZ arC , S - m f ° r PauI What the y ^r Peter. If the 

12 e™ T 15 rCaIIy What We had assu ™d, Peter can 

ThuJZl S 1S r rSiStenCe ° f relation * by crediting Paul with 

Wntfen , T ^ **» ™ ^ « in the 

put dow? C hC 10 rCCk0n ° therwise ' he would n0t 

Zt 72 l m hlS . mathemat ical representation of the world 

Z the el^ m ° tl0n diSC ° VerS am ° n S a11 Phenomena-includ- 
r P ft w„ • C L troma g netlc -the same relations as Peter does at 

maintained for pLl ih' I"' Why *' rdl " iOM 

Paul a, ,h»J J must ,be y be recorded by Peter for 

consequent "f t, * Same ri S ht as p ««' But it is a mere 

norr™^*L7f C,P 0„° c ? ,h \ he nOKS fa " ,iS ^ 
and Paul U « i u CC a S ain ' he becomes the referrer 

Paul ls only the referent. Since this is the case, Paul's 



time is a hundred times slower than Peter's. But it is at- 
tributed, not lived time. The time lived by Paul would be the 
time of Paul referring and no longer referent— it would be 
exactly the time that Peter just found. 

We always come back, then, to the same point: there is a 
single real time, and the others are imaginary. What, indeed, 
is a real time, if not a time lived or able to be lived? What is 
an unreal, auxiliary, imaginary time if not one that cannot 
actually be lived by anything or anyone? 

But we see the source of the confusion. We would formulate 
it as follows: the hypothesis of reciprocity can be expressed 
mathematically only in that of nonreciprocity, because to ex- 
press mathematically the freedom of choosing between two 
systems of axes is actually to choose one of them. 2 The faculty 
of choosing cannot be read in the choice we make by virtue 
of it. A system of axes, by the very fact that it has been 
adopted, becomes a privileged system. In its mathematical 
use, it is indistinguishable from an absolutely motionless sys- 
tem. That is why unilateral and bilateral relativity are mathe- 
matically equivalent, at least in the case at hand. The dif- 
ference exists here only for the philosopher; it shows up only 

we ask ourselves what reality, that is, what perceived or 
perceptible thing, the two hypotheses imply. The older, that of 
the privileged system in a state of absolute rest, certainly ends 
U P by positing multiple real times. Peter, really motionless, 
lives a certain duration; Paul, really in motion, would live 
a slower duration. But the other, that of reciprocity, im- 
plies that the slower duration must be attributed by Peter 
to Paul or by Paul to Peter depending upon whether Peter 
or Paul is the referrer or the referent. Their situations are 
identical; they live one and the same time but attribute 
differing times to each other and thus imply, in accord with 
th e rules of perspective, that the physics of an imaginary 
observer in motion must be the same as that of a real observer 
at rest. Hence, the hypothesis of reciprocity gives us at least 

2 What is, of course, always alone in question is the special theory of 


as much reason for believing in a single time as does common 
sense; the paradoxical idea of multiple times asserts itself only 
in the theory of the privileged system. But, once more, we 
can express ourselves mathematically only in the theory of the 
pnvileged system, even when we have begun by granting 
reciprocity; and the physicist, feeling free of the theory of 
reciprocity once he has done it homage by freely choosing his 
system of reference, surrenders it to the philosopher and hence- 
forward expresses himself in the language of the privileged 
system Paul will enter the projectile, believing in this physics. 
He will come to realize on the way that philosophy was rights 
What has helped foster the illusion is that the special theory 
ot relativity makes the precise claim of seeking for things 
a representation independent of the system of reference." It 

v P IrT h l t J le ° ry °u P^^ 1 sea kd ^ a projectile, and living only two 
1 Tl 1 ^ tW ° . ndred yCarS r ° U ^ on earth wa * «=t forth by Langevin 
Zo , f 'V 0 ,? 1 6 C ° ngreSS ° f Bol °S na in 1911 - " » widely known and 
TZt Y ? y ' U 18 f ° Und in J ean Becquercl'. important work, Le 
e. ae, m2l p T ^ la tMOrie ^ U gravitation (Paris: Gauthier-Villan 

JtZl wT ly ^ Stand P° int ° f Phy^cs, it raises certain difficulties, 
SanZ T " C here / eall y no lon S er in special relativity. As soon as speed 

SXanSit'y G " aCCderati0n ^ deali "^ WUh a P r ° bleffl 

do! U and n r y CaS6 ' S ° 1Uti0n given above completely removes the para- 
2 ^volume ^ WU ^ Pr ° blem - See the Appendixes at the end of 

Coliesfof R!!,° PPOrt u Unity t0 My that k was Langevin's address to the 
^^ ot^TJto^ drCW a " enti0n t0 EinStdn ' S ^ 
the works znZ^L^ZT ^ ^ " ^ " 

CongrTof Boll hiS ? T * ^ general S ath «ing of the Philosophic 
tZ Trespace Td °" ^ ^ He later P ub " shed il as " L ' EV ° 1U ' 
XIX (19H), 455-466 ] temPS ReVU€ de Meta P h y si 1 ue et de Morale, 

coZ:z e X*T£! r selve : to spedai m * *™ we are 

tend not to close any ys « oTtT^ ^ * ^ 

struction of an • Y reference, to proceed as for the con- 

cons"n7ek men "ri ge0metry Whh ° Ut —^ate axes, to use only 
consider is gene^y " * ^ «™ ^ the constancy that we actually 

themselves subordinate o £ Z * ^ e,ementS Wnkh 

to the choice of a system of reference. 


therefore seems to forbid the physicist to place himself at a par- 
ticular point of view. But there is an important distinction to 
be made here. Without doubt, the theoretician of relativity 
intends to give the laws of nature an expression that keeps its 
form in any system of reference to which events are referred. 
But this merely means that, placing himself, like any physicist, 
at a certain point of view, necessarily adopting a certain sys- 
tem of reference and thus noting down certain magnitudes, 
he establishes among these magnitudes relations that must be 
kept invariable among the new magnitudes he encounters 
should he adopt a new system of reference. It is precisely be- 
cause his method of inquiry and ways of notation assure him 
of an equivalence among all the representations of the uni- 
verse taken from every point of view that he has the absolute 
nght (ill assured in the old physics) to adhere to his personal 
point of view and to refer everything to his own system of 
reference. 5 But he is obliged to cling to this system generally. 
To this system the philosopher as well must therefore cling 
when he wishes to distinguish the real from the imaginary. 
The real is that which is measured by a real physicist, and the 
unaginary, that which is represented in the mind of the real 
Physicist as measured by imaginary physicists. But we shall 
return to this point in due course. For the moment, let us 
Point out another source of illusion, even less apparent than 
the first. 

The physicist Peter grants as a matter of course (this is only 
an °pinion, for it cannot be proven) that there are other con- 
sciousnesses like his, spread across the face of the earth, pos- 
sibly even at every point in the universe. It therefore makes 
no difference that Paul, John, and James are in motion with 
respect to him: he sees them as humans who think and feel 
as he does. This is because he is a man first and a physicist 

5 In his charming little book on the theory of relativity (The General 
Z'"" ple °f Relativity [London: MacMillan and Co., Ltd., 1920]). H. 
"don Can maintains that this theory implies an idealist conception of 
e un 'verse. We would not go that far; but we believe that it would 
artamly be necessary to orient this physics in an idealist direction if we 
ed t0 S ive »t the force of a philosophy. 



afterward. But when he thinks of Paul, John, and James as 
beings like himself, endowed with consciousnesses like his, he 
really forgets his physics or takes advantage of the license it 
grants him to speak in daily life like the common run of mor- 
tals. As a physicist, he is inside the system in which he makes 
his measurements and to which he refers everything. Men 
attached to the same system, and therefore conscious like him, 
will be physicists like him; they in fact work up, out of the 
same figures, the same world picture taken from the same 
point of view; they too are referrers. But the other men will 
be no more than referents; for the physicist, they can now 
be nothing but empty puppets. If Peter were to concede them 
feeling, he would at once lose his own; they would have 
changed from referents to referrers; they would be physicists 
and Peter would, in turn, have to become a puppet. This 
leaving-and-entering of consciousness, it might be added, ob- 
viously does not begin until we turn our attention to physics, 
because it is then clearly necessary to choose a system of refer- 
ence. Outside of that, the men remain as they are, one group 
like the other. There is no longer any reason for their not 
living the same duration and evolving in the same time. The 
plurality of times looms up at the precise moment when there 
is no more than one man or group to live time. Only that 
time then becomes real: it is the real time of a moment ago, 
but cornered by the man or group that has been given the 
status of physicist. All other men, having become marionettes 
from that moment on, henceforward evolve in times that the 
physicist imagines, which can no longer be real time, being 
neither lived nor able to be lived. Since they are imaginary, 
we can, of course, imagine as many of them as we like. 
What we are now going to add will seem paradoxical, yet it 

I™ -H ain T?' ThC idea ° f a reaI «™ common to two sys- 
tems, identical for S and S', asserts itself with greater force 
n the hypothecs of the plurality of mathematical times than 
nnf^r 1 V CCCpted ******** of a mathematical time, 

relativit U ? ,V Tc / **' " ^ h ^ thesis ° ther than that ° f 
relativity, S and S' are not strictly interchangeable; they occupy 


different positions with respect to some privileged system; and 
even if we have begun by making one a duplicate of the other, 
we see them immediately differing from one another by the sole 
fact of not maintaining the same relation to the central system. 
No matter how much we then attribute the same mathematical 
time to them, as had always been done before Lorentz and 
Einstein, it is impossible to demonstrate strictly that observers 
respectively placed in the two systems live the same inner dura- 
tion and that the two systems therefore have the same real 
time; it is, then, even very difficult to define this identity of 
duration with precision; all we can say is that we see no reason 
why an observer transferring from one system to another 
should not react the same way psychologically, live the same 
inner duration, for supposedly equal parts of the same mathe- * 
matical, universal time. This is sensible reasoning, to which >jj' 
nothing conclusive is opposed, but it is lacking in rigor and ; I 

precision. On the other hand, the hypothesis of relativity con- „i 
sists, in essence, of rejecting the privileged system; S and S' 
must therefore be regarded, while we are considering them, as i J 

strictly interchangeable if we have begun by making one the j; J 

duplicate of the other. But, in that event, the two people in 
s and S' can be led mentally to coincide, like two equal 3* 1 
superimposed shapes; they will have to coincide not only with ~\Z 
respect to the different modes of quantity but even, if I may ^ 
so express myself, in respect to quality, for their inner lives C 
have become indistinguishable, quite like their measurable J% 
features: the two systems steadfastly remain what they were £ 
at the moment we propounded them, duplicates of one an- i. 
other, while outside the hypothesis of relativity they were no ! " 

lo nger entirely so the moment after, when we left them to 
*eir fate. But we shall not labor the point. Let us simply say 
uiat the two observers in S and S' live exactly the same dura- 
tl0n ar »d that the two systems thus have the same real time. 

Is this still the case for every system in the universe? We 
assigned S' any velocity; we can then repeat for every S" 
system what we said about S'; the observer we attach to it will 
Ilve the same duration in it as in S. At most, it will be ob- 

b i 


jected that the reciprocal displacement of S" and S is not the 
same as that of S' and S, and that, consequently, when we 
immobilize S into a system of reference in the first case, we are 
not doing strictly the same thing as in the second. The dura- 
tion of the observer in motionless S, when S' is the system that 
we are referring to S, would not then necessarily be identical 
with that of this same observer when the system referred to 
S is S"; there would be, as it were, different intensities of 
immobility in keeping with the greater or lesser speed of the 
reciprocal displacement of the two systems before one of them, 
suddenly elevated to a system of reference, had been mentally 
immobilized. We do not think anyone wants to go that far. 
But, even then, we would simply adopt the position we usually 
take when we parade an imaginary observer across the world, 
judging it right to attribute the same duration to him every- 
where. We mean that we see no reason to believe the opposite; 
when things look one way, it is up to anyone who calls them 
illusory to prove them so. Now, the idea of assuming a plural- 
ity of mathematical times had never occurred before the 
theory of relativity; it is therefore to it alone that we would 
refer in order to cast doubt upon the unity of time. And we 
have just seen that in the only completely clear and precise 
case of two systems S and S' moving with respect to one an- 
other, the theory of relativity would end by supporting the 
unity of real time more rigorously than we do ordinarily. It 
permits defining and almost demonstrating this identity, in- 
stead of confining us to the vague and merely plausible asser- 
tion with which we are generally content. We conclude that, 
as far as the universality of real time is concerned, the theory 
ot relativity does not shake the accepted belief and tends 
rather to strengthen it. 

Let us now proceed to the second point, the breakup of 
simultaneities. But let us first recall in a few words what we 
said about intuitive simultaneity, the one we could call real 
and lived. Einstein necessarily acknowledges it, since, through 
it, he notes the time of an event. We may confer upon simul- 
taneity the most learned of definitions, saying that it is an iden- 


tity between the readings of clocks synchronized through an 
exchange of optical signals, and concluding that simultaneity 
is relative to the synchronizing. It is nonetheless true that we 
compare clocks in order to determine the time of events; but 
the simultaneity of an event with the clock reading that gives 
us its time does not follow from any synchronizing of events 
with clocks, it is absolute. 6 If it did not exist, if simultaneity 
were only correspondence between clock readings, if it were 
not also, and before all else, correspondence between a clock 
reading and an event, we would not build clocks, or no one 
would buy them. For we buy them only to find out what time 
it is. But "to find out what time it is" is to note the simul- 
taneity of an event, of a moment of our life or of the outside 
world, with a clock reading; it is not, in general, to record 
a simultaneity between clock readings. Hence, it is impossible 
°r the theoretician of relativity not to acknowledge intuitive 
simultaneity.* He makes use of this simultaneity in the very 

6I t is lacking in precision, to be sure. But when we fix this point 
^rough laboratory experiments, when we measure the "delay" caused by 

e psychological establishment of a simultaneity, it is to intuitive simul- 
itiil^ that - W mUSt Sti11 haVe recourse in order t0 criticize it. In the 
cession" 31 ^ 18 ' EVerythin S rests u P on intuitions of simultaneity and sue- 

ciDl n h may -' ° f course ' be teru pted to raise the objection that, in prin- 
with' 6 " n ° simultaneit y at a distance, however small the distance, 
cm, m 3 s y nchroni zing of clocks. One may reason as follows: "Let us 
it is' ^ y ° Ur intuitive ' simultaneity between two events A and B. Either 
over * me ^ ely a PP roxi mate simultaneity, the approximation being, more- 
events SUffiCIent considerin g the enormously greater distance separating the 
else it am ° ng which y° u ar e going to establish a 'learned' simultaneity; or 
aware "f * Per£ect simulta neity, but in that case, you are, without being 
synch ° • ° nly ascertainin g a " identity of readings between the two 
virtua r ° mzed microbial clocks of which you spoke earlier, clocks that exist 

have* y 31 A 3nd B ' If you alle S e that y our microbes posted at A and B 
We W o eC ,° Urse t0 'intuitive' simultaneity for the reading of their apparatus, 
'"bmi" 1 h . repeat our ar gument by this time imagining submicrobes and 
Would C fi° rf 1 ° l0CkS ' ln short ' the im P recision alwa y s diminishing, we 
de Pe * in the final reckoning, a system of learned simultaneities in- 
Proxi m ent ° f intuitive simultaneities; the latter are only confused, ap- 
ate ' Provisory visions of the former." But this argument runs 



synchronization of the two clocks through optical signals, and 
he makes use of it three times, for he must note: (1) the optical 
signal's moment of departure, (2) the moment of its arrival, 
(3) that of its return. Now, it is easy to see that the other 
simultaneity, the one that depends upon a synchronizing of 
clocks carried out through an exchange of signals is still called 
simultaneity only because we believe we can convert it into 
intuitive simultaneity. 8 The one who synchronizes the clocks 
necessarily takes them to be inside his system; as this system 
is his system of reference, he deems it motionless. For him, 
therefore, the signals exchanged between two clocks at a dis- 
tance from one another make the same trip leaving as return- 
ing. Were he to place himself at any point equidistant from the 
two clocks, and were his eyes sharp enough, he would grasp 
the readings of the two optically synchronized clocks in one 
instantaneous intuition and would at that moment see them 
pointing to the same time. To him learned simultaneity there- 
fore always appears able to be converted into intuitive simul- 
taneity, which is why he calls it simultaneity. 

This being granted, let us consider two systems S and S' in 
motion with respect to one another. Let us first take S as our 
system of reference. By that very act we immobilize it. Clocks 
have been synchronized in it, as in every system, through an 
exchange of optical signals. As in every synchronizing, it has 

counter to the very principle of the theory of relativity, which is never to 
assume anything more than has actually been found out and actually 
ascertained by measurement. It would be to postulate that anterior to our 
human knowledge, which is in a perpetual becoming, there is a knowledge 
in full, given in eternity in one piece and mingling with reality itself-we 
would be limited to acquiring the latter, bit by bit. Such was the ruling 
idea of Greek metaphysics, an idea revived by modern philosophy and, 
it must be added, natural to our human understanding. I do not mind 
our concurring in it, but we must not forget that it is a metaphysic, and 
a metaphysic based upon principles that have nothing in common with 
those of relativity. 

8 We showed further back (pp. 55-56) and have just repeated that one 
cannot make a radical distinction between local simultaneity and simul- 
taneity at a distance. There is always a distance which, however small it 
may be for us, will appear enormous to a microbe-builder of micro- 
scopical clocks. 



then been assumed that the exchanged signals made the same 
trip out and back. Indeed, they really do so, since the system 
is motionless. If we designate C m and C n as the points where 
the two clocks are, an observer inside the system, choosing any 
point equidistant from C m and C n will be able, if he has sharp 
enough eyes, to embrace from there, in a single act of instanta- 
neous vision, any two events occurring at points C m and C n 
respectively when these two clocks show the same time. Spe- 
cifically, he will embrace in this instantaneous perception the 
two concordant readings on the two clocks— readings that are 
also themselves events. Every simultaneity indicated by clocks 
will then be able to be converted into intuitive simultaneity 
inside the system. 

Let us then consider system S'. It is clear that the same will * 
happen for an observer inside this system. This observer takes t i 

s ' as his system of reference. He therefore renders it motion- ) $ 

less. The optical signals by means of which he synchronizes his ^ 1 

clocks then make the same trip out and back. Hence, when two ^ „ 

of his clocks show the same time, the simultaneity they in- ;! y 

dicate could be lived and become intuitive. i< « 

Thus, there is nothing artificial or conventional in simul- ^ 
taneity whether we apprehend it in one or the other of the .1-1 
two systems. j 

B ut let us now see how one of the two observers, the one in CJ 
S > judges what is happening in 5'. For him, the S' system is in 
m °tion and, as a consequence, optical signals exchanged be- t ; 
tw een its two clocks do not make the same trip out and back, £ 
as an observer attached to the system would believe (except, ^ i. 

° f course . in the special case of two clocks lying in the same i * 

Plane perpendicular to the system's direction of motion). 
Therefore, i n his eyes, the synchronizing of the two clocks has 
een Performed in such a way that they give the same reading 
w hen there is no simultaneity, but succession. Only, let us 
re mark that he is thus adopting an entirely coventional defini- 
tl0n of succession, and therefore of simultaneity as well. He 
a grees to call successive the concordant readings of clocks that 

1 have been synchronized under the conditions that he 



perceives in system S'— I mean so synchronized that an ob- 
server outside the system does not ascribe the same trip to the 
optical signal out and back. Why does he not define simul- 
taneity by the agreement between readings on clocks so syn- 
chronized that the outward and return journeys are the same 
for observers inside this system? The answer is that each of the 
two definitions is valid for each of the two observers and that 
this is precisely why the same events in system S' can be de- 
clared simultaneous or successive, according to whether they 
are envisaged from the point of view of S or S'. But it is easy 
to see that one of the two definitions is purely conventional, 
while the other is not. 

To verify this, we are going to come back to a hypothesis 
that we have already set forth. We shall assume that S' is a 
duplicate of system S, that the two systems are identical, that 
the same history unfolds within them. They are in a state of 
reciprocal movement, completely interchangeable; but one of 
them is adopted as a system of reference and is from then on 
deemed motionless; this will be S. The hypothesis that S' is a 
duplicate of S is not damaging to the generality of our demon- 
stration, since the alleged breakup of simultaneity into suc- 
cession, and into a succession more or less slow as the move- 
ment of the system becomes more or less rapid, depends only 
upon the system's speed, and not at all upon its content. This 
granted, it is clear that if events A, B, C, D of system S zee 
simultaneous for the observer in S, the identical events A', B', 
C, D' of system S' will also be simultaneous for the observer 
in S'. Now, will these two groups A, B, C, D and A', B' , C, V, 
each formed of events simultaneous for an observer inside the 
system be additionally simultaneous, that is, perceived as 
simultaneous by a supreme consciousness capable of instantly 
sympathizing or telepathically communicating with the two 
consciousnesses in S and S'} It is obvious that there is no objec- 
tion to this. Indeed, we can imagine, as just before, that the 
duplicate S' has broken away from S at a certain moment and 
is then obliged to return to it. We have demonstrated that 
the observers inside the two systems will have lived the same 



total duration. We can therefore divide this duration in both 
systems into a like number of slices such that each one of them 
is equal to the corresponding slice in the other system. If the 
moment M when the simultaneous events A, B, C, D, occur 
is found at the extremity of one of the slices (and this can 
always be arranged), the moment M' when the simultaneous 
events A', B', C , D' occur in system S' will be the extremity 
of the corresponding slice. Situated like M, inside an interval 
of duration whose ends coincide with those of the interval where 
M is found, it will necessarily be simultaneous with M. And 
consequently the two groups of simultaneous events A, B, C, D 
and A', B' , C, D' will really be simultaneous with each other. 
We can therefore continue to imagine, as in the past, in- 
stantaneous slices of a single time and absolute simultaneities 
of events. 

But, from the viewpoint of physics, the argument we have 
just advanced will be of no consequence. In physics, the prob- 
fem is, in effect, posed in the following manner: if S is at rest 
and S' in motion, why do experiments on the speed of light, 
carried out in S, give the same result in S'? And it is under- 
stood that only the physicist in system S exists as a physicist- 
*e one in system S' is merely imagined. Imagined by whom? 
Necessarily by the physicist in system S. The moment we make 
s our system of reference, it is from there, and from there only, 
wat a scientific world view is thenceforth possible. To keep 
observers in S and in S' conscious at one and the same time 
w ould be to sanction both systems' being given the status of 
systems of reference and ruled motionless together; but they 
ave been assumed in a state of reciprocal motion; at least 
° ne of the two must therefore be moving. To be sure, we shall 

hH- e men in the movin s one ; but the y wil1 have m° mentaril y 

a cheated their consciousness or, at least, their faculties of 
° servation; they will retain, in the eyes of the single physicist, 
° n ty the physical side of their person as long as it is a question 
v P h y si «- From here on, our argument gives way, for it in- 
Wved the existence of equally real men, similarly conscious, 
J°ying the same rights in both system S' and in system S. 


It can no longer be a question of more than one group of 
men— real, conscious physicists— those in the system of refer- 
ence. The others would indeed be hollow puppets or else they 
would be only virtual physicists, merely conceived in the mind 
of the physicist in S. How will the latter picture them? He will 
imagine them, as before, experimenting with the speed of 
light, but no longer with a single clock, no longer with a mir- 
ror that reflects the beam of light and doubles its journey; 
there is now a single journey and two clocks respectively 
located at the points of departure and arrival. He will then 
have to explain how these imagined physicists would find the 
same speed for light as he, the real physicist, if this entirely 
theoretical experiment were to become realizable in practice. 
Now, as he sees it, light moves at a slower speed for system S' 
(the conditions of the experiment being those we indicated a 
while back); but also, since the clocks in S' have been so 
synchronized as to mark simultaneities where he perceives suc- 
cessions, things will work out in such a way that the real ex- 
periment m S and the merely imagined experiment in S' will 
give the same figure for the speed of light. This is why our ob- 
server m 5 holds to the definition of simultaneity that makes it 
depend upon the synchronization of clocks. That does not 
prevent the two systems, S' as well as S, from harboring real, 
iived simultaneities, not governed by clock synchronizations. 

we must therefore make a distinction between two kinds 
ot simultaneity and succession. The first is inside events, a part 
ot toeir materiality, proceeding from them. The other is merely 
laid down over them by an observer outside the system. The 
first says something about the system itself; it is absolute. The 
dTZ? 15 Ch f geable ' relati ^, imaginary; it turns upon the 
It,? 6 ; <fn g Wkh s P eed > be tween this system's im- 
mobility for nself and its mobility with respect to another; 

ces-L xk 3 ? arent incurvation from simultaneity into sue- 
to an .'a ^simultaneity and the first succession belong 
of them ^f* f thingS: the second ' to observer's image 
±e s^edt n in K "? mirr ° rS that distort *e more, the greater 
the speed attributed to the system. The incurvation of W 



taneity into succession is, moreover, just what is required for 
the laws of physics, particularly those of electromagnetism, to 
be the same for the observer within the system who is located 
in the absolute, as it were, and the observer outside, whose 
relation to the system can vary indefinitely. 

I am in system S', which is assumed to be motionless. I note 
intuitive simultaneities there between two spatially separated 
events, O' and A', having taken up a position equidistant from 
both. Now, since the system is motionless, a light ray that 
leaves and returns between points O' and A' makes the same 
trip out and back; if I then work the synchronizing of the two 
clocks, respectively located at O' and A', under the assumption 
that the outward and return passages P and £> are equal, I am 
m the right. Thus I have two ways of recognizing simultaneity 
at this point: the one, intuitive, by encompassing what occurs 
at 0' and A' in an act of instantaneous vision; the other, 
derivative, by consulting the clocks; and the two results agree. 
I now assume that, nothing of what is happening in system S 
having changed, P no longer seems equal to A. This is what 
happens when an observer outside S' perceives this system in 
motion. Are all the former simultaneities 9 going to become 
successions for this observer? Yes, by convention, if we agree to 
translate all the temporal relations of all the events in the sys- 
tem into a language such as makes it necessary to change their 
expression in accordance with whether P appears equal or un- 
gual to Q. This is what we do in the theory of relativity. 
' a re] ativist physicist, after having been inside the system and 
Perceived P equal to leave it; entering an indefinite number 
of systems assumed motionless by turns and with respect to 
which S' would then be found endowed with increasing speeds, 
see the inequality between P and Q_ increasing. I then declare 
at the events that were simultaneous before are becoming 
Recessive, and that their temporal separation is increasing. 
. ut we have here only a convention, a necessary convention, 
U must °e added, if I wish to preserve the integrity of physical 

sale XCePti ° n " made ' of course ' of those relatin S to events located in the 
e plane perpendicular to the direction of motion. 



laws. For it just so happens that these laws, including those of 
electromagnetism, have been formulated under the assumption 
that physical simultaneity and succession are defined by the 
apparent equality or inequality of the P and Q journeys. In 
stating that succession and simultaneity depend upon one's 
point of view, we are doing nothing more than giving expres- 
sion to this assumption, recalling this definition. Are we deal- 
ing with real simultaneity and succession? We are dealing 
with reality, if we agree to call any convention representative 
of the real once it has been adopted for the mathematical 
expression of physical facts. So be it; but then let us no longer 
speak of time; let us say that we are dealing with a succession 
and simultaneity that have no connection with duration; for, 
by virtue of a prior and universally accepted convention, there 
is no time without a before and an after verified or verifiable 
by a consciousness that compares one with the other, were this 
consciousness only an infinitesimal consciousness coextensive 
with the interval between two infinitely adjacent instants. 
If you define reality by mathematical convention, you get a 
conventional reality. But actual reality is what is, or could be, 
perceived. But, once again, outside of this double journey Pg 
which changes in aspect according to whether the observer is 
inside or outside the system, everything perceived and per- 
ceptible in S' remains as it is. This means that it does not 
matter whether S' is considered at rest or in motion-real simul- 
taneity remains real simultaneity; and succession, succession. 

When you kept S' motionless and consequently placed your- 
self inside this system, learned simultaneity (the one we de- 
duced from the agreement between optically synchronized 
clocks) coincided with intuitive or innate simultaneity; and it 
is only because it was of use to you in recognizing this innate 
simultaneity, because it was its token, because it was con- 
vertible into intuitive simultaneity, that you called it simul- 
taneity. Now, S> being ruled in motion, the two kinds of simul- 
taneity no longer coincide; all that was innate simultaneity 
remains innate simultaneity; but the faster the system's speed, 
the greater grows the inequality between the P and Q journeys, 



although it was by their equality that the learned simultaneity 
was defined. What ought you to do if you felt sorry for the 
poor philosopher, condemned to a tete-a-tete with reality, ac- 
quainted with it alone? You would give another name to the 
learned simultaneity, at least when you talk philosophy. You 
would invent another word for it, any word, but you would 
not call it simultaneity, for it owes this name solely to the 
fact that it betokened the presence of a natural, intuitive, real 
simultaneity in S' assumed motionless, and that we can now 
believe that it still denotes this presence. You yourself, more- 
over, keep admitting the legitimacy of this original meaning 
of the word, at the same time as its primacy; for when S' seems 
to you to be in motion, when, speaking of the agreement of its 
clocks, you seem no longer to be thinking of learned simul- 
taneity, you keep appealing to the other, the real one, through 
your establishment of a "simultaneity" between a clock read- 
ing and an "adjacent" event (adjacent for you, a man, but 
vastly separated for a discerning microbe-scientist). Neverthe- 
less, you hold on to the word. Indeed, through this word com- 
mon to both cases and working magically (does not science 
act upon us like ancient magic?) you perform a transfusion of 
reality from one simultaneity to the other, from innate to 
'earned simultaneity. The passing from stability to mobility 
having doubled the meaning of the word, you slip all the 
materiality and solidity of the first meaning into the second. 
I would say that instead of forewarning the philosopher against 
Ais error, you want to draw him into it, did I not realize the 
advantage you derive, as a physicist, from using the word 
simultaneity in both senses: you remind yourself in this way 
Aat learned simultaneity began as innate simultaneity and can 
always turn into it again should thought immobilize the 
system anew. 

Fr om the point of view which we called that of unilateral 
relativity, there is an absolute time and an absolute clock-time, 
th e time and clock-time of the observer located in the privi- 
leged system S. Let us assume once more that S', having at first 
coincided with S, has then separated from it by way of doubling. 



We can say that the clocks in S', which continue to be syn- 
chronized in the same way, by optical signals, show the same 
clock-time when they ought to show different clock-times; 
they note simultaneity in cases where there is actually succes- 
sion. If, then, we take the position of unilateral relativity, we 
shall have to admit that the simultaneities in 5 break up in 
its duplicate S' by sole virtue of the motion that causes S' to 
leave S. To the observer in S' they appear to be retained, but 
they have become successions. On the other hand, in Einstein's 
theory, there is no privileged system; the relativity is bilateral; 
everything is reciprocal; the observer in S is as much in the 
right in seeing succession in S' as is the observer in S' in seeing 
simultaneity there. But what are also in question are the suc- 
cessions and simultaneities defined solely by the appearance 
assumed by the two journeys P and Q. The observer in S' is 
not mistaken, since, for him, P is equal to Q: the observer in 
5 is no more mistaken, since, for him, the P and Q of system 
S' are unequal. But, unconsciously, after accepting the theory 
of double relativity, we revert to that of single relativity, first, 
because they are mathematically equivalent, then, because it 
s very difficult not to imagine according to the latter when we 
hink according to the former. We then act as if-the two 
passages P and £> appearing unequal when the observer is out- 
side S'-the observer inside S' were mistaken in designating 
these passages as equal, as if events in the physical system S' 
had been broken up in actuality at the dissociation of the two 
systems, when it is merely the observer outside S' who rules 
them broken up in following his own definition of simul- 
taneity. We forget that simultaneity and succession have then 
become conventional, that they retain of the original simul- 
taneity and succession merely the property of corresponding 
to the equahty or inequality of the two journeys P and ft. 
It was then still a question of an equality and inequality 
found by an observer inside the system and therefore final and 

We shall easily be convinced that the confusing of the two 
viewpoints is natural and even inevitable, when reading cer- 



tain pages in Einstein himself. Not that Einstein was obliged 
to commit this error, but the distinction we have just drawn 
is of such a nature that the language of the physicist is hardly 
able to express it. It is, besides, of no importance to the 
physicist, since the two conceptions are conveyed in the same 
manner in mathematical terms. But it is the essential point 
for the philosopher, who will picture time altogether differ- 
ently according as he takes one position or the other. The 
pages that Einstein has devoted to the relativity of simultaneity 
in his book on The Theory of Special and General Relativity 
are instructive in this regard. We quote the heart of his 

M' >- >- 


M B 

Figure 3 

Suppose that an extremely long train moves on its track at a 
speed v, as shown in Figure 3. The passengers on this train will 
choose to consider it as their system of reference; they will refer 
every event to the train. Every event that takes place at a poini t on 
the track also takes place at a particular point on the train. The 
definition of simultaneity is the same with respect to the tram as 
w »h respect to the track. But the following question then arises: 
are two events (for example two flashes of lightening A and B) si- 
multaneous with respect to the track also simultaneous with respect 
t0 the train? We shall straightaway show that the answer is in the 
negative. In saying that the two flashes of lightning A and B are 
simultaneous with respect to the track, this is what we mean: the 
^ght rays emitted from points A and B will meet in the middle M 
of the distance AD measured along the track. But to the events A 
and B there also correspond points A and B on the train- Suppose 
J « the middle of the vector AB of the moving train This point 
^ certainly coincides with point M at the instant the Hash" ° r 
hghtriing occur (an instant recorded with respect to the track) du 
" then moves to the right on the diagram at speed v of the train. 


If an observer at AT on the train were not borne along at this speed, 
he would remain constantly at M, and the light rays emitted from 
points A and B would reach him simultaneously, that is, these rays 
would cross exactly upon him. But, in reality, he is traveling (with 
respect to the track) and is proceeding toward the light from B, 
while fleeing the light from A. The observer will therefore see the 
first sooner than the second. Observers who take the track as their 
system of reference conclude that the flash of lightning B has oc- 
curred before the flash of lightning A. We therefore arrive at the 
following basic fact. Events simultaneous with respect to the track 
are no longer so with respect to the train, and vice versa (relativity 
of simultaneity). Each system of reference has its own time; a time 
reading has meaning only if we indicate the system of comparison 
used for the measurement of time.™ 

This passage enables us to catch on the wing an ambiguity 
that has been the cause of a good many misunderstandings. 
To clear it up, we shall begin by drawing a more complete 
figure (Figure 4). Notice that Einstein has indicated the train's 



J L 

I i 

M^r B -<- 

Figure 4 


direction by arrows. We shall indicate the opposite direction 
oi the track by other arrows. For we must not forget that the 
tram and the track are in a state of reciprocal motion. To be 
sure, Hinstem does not forget this either when he refrains from 
drawing arrows along the track; he thereby indicates that he 
chooses the track as his system of reference. But the phi- 
losopher who wants to know what to believe regarding the 
nature of time, who wonders whether or not the track and the 

HrlT a S3me ^ time ~ that the same lived or livable 
time-the philosopher must always remember that he does not 

j ivouviere (P ans: Gauthier-Villars, 1921), pp. 21, 22. 



have to choose between the two systems; he will place a con- 
scious observer in both and will seek out the lived time of 
each. Let us therefore draw additional arrows. Let us now add 
two letters, A' and B', to mark the extremities of the train. By 
not giving them labels of their own, by leaving them with the 
letters A and B of the points on the earth with which they 
coincide, we would once again risk forgetting that both track 
and train are subject to the rule of complete reciprocity, and 
enjoy equal independence. Finally, we shall more generally 
call M' any point on the line A'B' which will be located with 
respect to B' and A' as M is with respect to A and B. So much 
for the Figure. 

Let us now emit our two flashes of lightning. The points 
from which they set out no more belong to the ground than 
to the train; the waves advance independently of the motion 
of their source. 

It then becomes evident at once that the two systems are 
interchangeable, and that exactly the same thing will occur at 
M' as at the corresponding point M. If M is the middle of AB, 
and if it is at M that we perceive a simultaneity on the track, 
it is at M', the middle of B'A', that we shall perceive this same 
simultaneity in the train. 

Accordingly, if we really cling to the perceived, to the lived, 
if we question a real observer on the train and on the track, 
we shall find that we are dealing with one and the same time- 
what is simultaneity with respect to the track is simultaneity 
with respect to the train. 

fi ut, in marking the double set of arrows, we have given up 
adopting a system of reference; we have mentally placed our- 
selves on the track and in the train at one and the same time; 
we have refused to turn physicist. We were not, in fact, looking 
fc * a mathematical representation of the universe; the latter 
must naturally be conceived from one point of view and con- 
form to the laws of mathematical perspective. We were asking 
ourselves what is real, that is, observed and actually recorded. 

On the other hand, for the physicist, there is what he him- 
self records-this, he notes as it is-and then there is what he 
records of another's possible recording; this he will transpose, 



lead around to his point of view, since every physical repre- 
sentation of the universe has to be referred to a system of refer- 
ence. But his notation of it will then no longer correspond to 
anything perceived or perceptible; it will therefore no longer 
be a notation of the real but of the symbolic. The physicist 
located in the train will therefore entertain a mathematical 
vision of the universe in which everything will be converted 
from perceived reality into useful scientific representation, 
except what relates to the train and the objects attached to it. 
The physicist on the track will entertain a mathematical vision 
of the universe in which everything will be similarly trans- 
posed, except what concerns the track and the objects bound 
to it. The magnitudes appearing in these two visions will be 
generally different, but, in both, certain relations among mag- 
nitudes, which we call the laws of nature, will remain the 
same, and this identity will precisely express the fact that the 
two representations are of one and the same thing, of a 
universe independent of our representation. 

What then does the physicist located at M see on the track? 
He records the simultaneity of the two flashes of lightning. Our 
physicist cannot be at point M' also. He can only say that he 
ideally sees the recording at M> of a nonsimultaneity between 
tne t wo flashes His mathematical representadon Qf the world 
will rest entirely on the fact that his adopted system of refer- 
ence is tied to the earth. Accordingly, the train moves; accord- 
ingly, we cannot grant the simultaneity of the two flashes of 
hghtmng at M>. The truth is that nothing has been recorded 
since, for that, a physicist at M' would be needed and the 
on y physicist in the world is, by hypothesis, at M. There is 
nothing more at M' than a certain notation carried out by 
the observer at M, a notation which is, indeed, that of a non- 
r 7, Y ' • lf W pref€r ' there is a ™™ly imagined physi- 

with ro™ 1 7 u ke Elnstei n. What is simultaneity 

Id he wn H " n0t S ° With res P"< ^0 the train." 

from Ae n r 7^' * adds ' " since * U P 

" e r: d o£ .. v wh w at°L the r k - ■ He Jouid - moreov r 

' What 1S simultaneity with respect to the 



train is not so with respect to the track, since physics is built 
up from the point of view of the train." And, finally, he would 
have to say: "A philosophy which assumes the viewpoints of 
both track and train, which then notes as simultaneity in the 
train what it notes as simultaneity on the track, no longer stands 
halfway between perceived reality and scientific construction; 
it is completely in the real, and is moreover, only completely 
appropriating Einstein's conception which is that of the re- 
ciprocity of motion. But that idea, as complete, is philosophical 
and no longer physical. To convey it in physicist's language, 
we must take the position of what we called the hypothesis of 
unilateral relativity. And as this language asserts itself, we 
do not perceive that we have for a moment adopted this hy- 
pothesis. We then speak of a multiplicity of times that are 
all on the same plane, all real, therefore, if one of them is 
real. But the truth is that the latter differs fundamentally 
from the others. It is real, because it is really lived by the 
Physicist. The others, merely thought of, are auxiliary, mathe- 
matical, symbolic." 

But, the ambiguity is so difficult to clear up that we can- 
not attack it from too many angles. Let us therefore consider 
(Figure 5) three points W, N', F in system S' so arranged on 
a straight line marking the direction of the system's motion 

Figure 5 


that N' is the same distance I from AT and P'. Let us imagine a 
person at N'. At each of the three points M', N', P' a series of 
events unfolds constituting the history of the place. At a parti- 
cular moment, the person at N' perceives a completely deter- 
minate event. But are the events contemporaneous with this 
one, occurring at N' and P', determinate as well? No, according 
to the theory of relativity. Depending upon the speed of system 
S', neither the same event at M' not at P' are contemporaneous 
with the event at N'. If, then, we regard the present of the per- 
son at N' as constituted, at a given moment, by all the simulta- 
neous events that come into being at that moment at all points 
in his system, only a fragment of it is determinate. This is the 
event occuring at point N' where the person happens to be. The 
rest will be indeterminate. The events at M' and P', which 
are also part of our person's present, will be this or that ac- 
cording as we attribute one speed or another to system S, ac- 
cording as we place him in this or that system of reference. 
Let us call its speed v. We know that when properly synchro- 
mzed clocks show the same time at the three points, and con- 
sequently, when there is simultaneity in system S', the observer 
m the S system of reference sees the clock at M' move ahead of 
and theclock at P' lag behind the one at N', both lead and lag 

being - seconds of system S'. Hence, for the observer outside 
the system, it is the past at M' and the future at P' that enter 
withm the present context of the observer at N'. What, at W 
and P , is part of the present of the observer at N' appears to 
this outside observer as being the farther back in the past his- 
tory of place N', the farther forward in the future history of 
place P the greater the system's speed. Let us then drop per- 
pendicular, tfff and P'K' to line M'P' in two opposite direc- 
tions, and let us suppose that all the events of the past history 
of place M are spaced out along M'H', all those of the future 

? a l7"l P u' al ° ng P ' K '- We can call "line of simul- 
" J* 6 f ^ ht line P**ing through point N>, joining the 
events E> and F' located, for the observer outside the system, 

an ^ time interval in the past of place AT and in the future 



of place F (the number designating seconds in system S'). 

This line, we see, keeps diverging from M'N'P' as the speed of 
the system increases. 

Here again the theory of relativity takes on, at first glance, 
a paradoxical appearance, striking the imagination. At once 
the idea comes to mind that if the gaze of our person at N' 
could instantly leap the space that separates him from F, he 
would perceive there a part of the future of that place, since 
it exists there, since it is a moment of that future which is 
simultaneous with this man's present. He would thus predict 
for an inhabitant of place F events that the latter will wit- 
ness. "To be sure," we tell ourselves, "instantaneous vision 
at a distance is not possible in actual fact; there is no speed 
greater than that of light. But an instantaneity of vision can 

°e imagined, and that is enough for the interval ^ of the fu- 
ture of place F rightfully to pre-exist in its present, to be 
preformed there and consequently predetermined." We shall 
see that this is a mirage. Unfortunately, the theoreticians of 
relativity have done nothing to dispel it. They have, on the 
contrary, seen fit to intensify it. The moment has not yet 
c °me for analyzing Minkowski's conception of space-time, 
adopted by Einstein. It has been expressed in a very ingenious 
schema into which, if we were not on our guard, we would 
r "k reading what we have just pointed out, into which, in- 
deed, Minkowski himself and his followers have actually 
f ead it. Without as yet applying ourselves to this schema (it 
w °uld call for a whole series of explanations which we may 
b ypass for the moment), let us convey Minkowski's thought, 
Usin 8 the simple figure we just drew. 

K we examine E'N'F, our line of simultaneity, we see that, 
at first merged with M'N'F, it gradually diverges as the speed 
v °f system S' increases with respect to the system of reference 
s _ But it win not diverge indefinitely. We know, in fact, that 
***** is no speed greater than that of light. Hence, the dis- 
hes M'E' and FF, equal to ^, cannot exceed I Let us grant 


them this length. We shall have, we are told, beyond E' in 
the direction of E'H', a region of absolute past, and beyond 
F' in the direction F'K', a region of absolute future; nothing 
of this past or future can be a part of the present of the ob- 
server at N'. But, in return, none of the moments in interval 
M'E' or P'F' is either absolutely before or after the one pass- 
ing at N'; all these successive moments of the past and future 
will be contemporaneous with the event at N', if we like; it 
will suffice to attribute the appropriate speed to system S', 
that is, in consequence, to choose the system of reference. Any- 
thing that has occurred at AF in an elapsed interval ^ , any- 
thing that will take place at P' in an interval 1 yet to elapse 

can enter the partly indeterminate present of the observer at 
AT'-the speed of the system will decide what will enter. 

The theoreticians of relativity, it must be added, have im- 
plicitly admitted that, if the observer at N' had the gift of 
instantaneous vision at a distance, he would perceive as pres- 
ent at P' what is going to happen there, since they have taken 
care to reassure us about the consequences of such a state of af- 
fairs." In actual fact, they point out, the observer at N' will 
never make use of this immanence, in his present, of what is 
hi the past at M' for the observer at W or of what is in the 
future at P' for the observer at P>; never will he profit from 
it or cause the inhabitants of W and P' to rue it; for no mes- 
sage can be transmitted, no causality exercised, at a speed 
greater than that of light; so that the person at N' could nei- 
ther be informed of a future of P' that is nevertheless a part 
ot his present, nor influence the future in any way; that future 
can with impunity be included in the present of the person 
at N; practically, it remains nonexistent for him. 

BulZnf, rT"'. 566 P - LangCVin ' " Le tem P s ' l'«pace et la causality" 
vl tC 1mnfaise de PhUosophie (1912); and Sir Arthur 

SnT "ttTp ESpaC€ ' tempS £t (Space Time, and Gravi- 

tanon), trans. J. Rossxgnol (Paris: J. Hermann, 1921), pp. 61-66. 



Let us see if this is not a mirage. We shall return to a sup- 
position which we have already made. According to the theory 
of relativity, the temporal relations among events unfolding 
in a system depend solely upon the speed of that system, not 
upon the nature of those events. The relations will therefore 
remain the same if we make S' a double of 5, unfolding the 
same history as S and having begun by coinciding with it. 
This assumption will greatly facilitate matters, and it will in 
no way detract from the generality of our demonstration. 

Accordingly, there is in system S a line MNP from which 
the line M'N'P' has parted by way of doubling, at the moment 
S' split from S. By hypothesis, an observer located at AF and 
one at M, being at two corresponding places in two identical 
systems, each witnesses the same history of the place, the same 
march of events. The same holds for the two observers at N 
and N' and for those at P and P', as long as each of them con- 
siders only the place where he is. With this everyone agrees. 
Now, we are going to pay particular attention to the two ob- 
servers at N and N', since what is in question is the simul- 
taneity with what is happening at these midpoints. 12 

For the observer at N, that which at M and P is simultan- 
eous with his present is fully determinate, since the system is 
motionless by hypothesis. 

As for the observer at N', that which at M' and F was simul- 
taneous in his present, when his system S' coincided with S, 

12 To simplify the argument, we shall assume in all that follows that 
the same event is in the act of being performed at points N and N' in the 
tWl n systems S and S'. In other words, we shall look at N and N' at the 
P r «ise instant of the dissociation of the two systems, allowing system S' 
10 squire its speed v instantly, in a sudden spurt, without passing 
through intermediate speeds. Upon this event constituting the common 
P r *ent of the two people at N and N>, we then fix our attention. When 
We shall state that we are increasing speed v, we shall mean that we are 
PWing things back in place, making the two systems coincide again, that, 
consequently, we are having the persons at N and N' witness the same 
^ and that we are then dissociating the two systems by imparting to 

• a gain instantly, a speed greater than the one before. 



was equally determinate. They were the same two events 
which, at M and P, were simultaneous in the present of N. 

S' now shifts with respect to S and acquires increasing 
speeds. But for the observer at N', inside S', this system is 
motionless. The two systems S and S' are in a state of complete 
reciprocity; it is for the convenience of study, to erect a phys- 
ics, that we have immobilized one or the other into a system of 
reference. All that a real, flesh-and-blood observer observes at 
N, all that he would instantaneously, telepathically observe 
at no matter how remote a point in his system would be 
identically perceived by a real flesh-and-blood observer located 
at N' in S'. Hence, that portion of the history of places W 
and P' which really enters the present of the observer at N' 
for him, what he would perceive at M' and P' if he had the 
gift of instantaneous vision at a distance, is determinate and 
unchanging, whatever the speed of S' in the eyes of the ob- 
server inside system S. It is the same portion that the observer 
at N would perceive at M and P. 

Let us add that the clocks of S' run for the observer at N' 
absolutely like those of S for the observer at N, since S and S' 
are in a state of reciprocal motion and, consequently, are 
interchangeable. When the clocks located at M, N, P, and 
which are optically synchronized, show the same time and 
when there is then, by relativist definition, simultaneity among 
the events occurring at these points, the same is true for the 
corresponding clocks in S'; and there is then, still by defini- 
tion, simultaneity among the events occurring at M', N', Pr- 
events respectively identical with the former ones. 

But, as soon as I have immobilized S into a system of ref- 
erence, here is what happens. In system S turned motionless, 
whose clocks we had optically synchronized, as we always do 
under the assumption of the system's immobility, simultaneity 
is something absolute; I mean that its clocks having been syn- 
chronized by observers necessarily in the system, on the as- 
sumption that optical signals between two points N and P 
make the same trip out and back, this assumption becomes 



definitive, is consolidated by the fact that S has been chosen 
as system of reference and definitively immobilized. 

But, by that very fact, S' is in motion; and the observer in 
S then notices that the optical signals between the two clocks 
at N' and P' (which the observer in S' supposed and still sup 
poses to be making the same trip out and back) now cover 
unequal distances, the inequality growing with every increase 
in the speed of S'. By virtue of his definition, then (for we are 
assuming the observer in S to be a relativist), the clocks that 
show the same time in system S' do not, in his eyes, underline 
contemporaneous events. There certainly are events that are 
contemporaneous for him in his system, as also there are 
events that are contemporaneous for the observer at N' in his 
own system. But to the observer at N they appear as successive 
in system S', or rather, they appear as having to be noted down 
as successive, by reason of his definition of simultaneity. 

Then, as the speed of S' increases, the observer at N drives 
farther into the past of point AT and projects farther into the 
future of point f-by the numbers he assigns them-events, 
occurring at these points, which are contemporaneous both 
for him in his own system and also for an observer located 
m system S'. For this last observer, it must be added, there is 
no further question of a flesh-and-blood existence; he has been 
surreptitiously drained of his content, in any case, of his con- 
sciousness; from observer he has become simply observed, 
Sln ce it is the observer in N who has been given the status 
of Physicist-builder of all science. Consequently, I repeat, as 
v increases, our physicist notes as pushed back ever farther 
"Jto the past of place M', advanced ever more into the future 
°f place P>, the always identical event which, whether it be 
« M' or F, i s part of the really CO nscious present of an ob- 
server at N', and consequently part of his own. There are not, 
toerefore, different events at place P' which enter by turns, for 
^easing speeds of the system, into the real present of the 
observer at N'. But the same event of place F, which is part 
2 ^ present of the observer at N', under the assumption of 
™ e system's immobility, is noted by the observer at N as be- 



longing to a future ever more remote from the observer at 
N', as the speed of the mobilized system S' increases. If the 
observer at N did not so note, it must be added, his physical 
conception of the universe would become incoherent, for his 
written measurements of phenomena occurring in a system 
would express laws that he would have to vary with the sys- 
tem's speed; thus, a system identical with his, whose every 
point would have identically the same history as the corre- 
sponding point in his, would not be governed by the same 
physics (at least in what concerns electromagnetism). But 
then, in noting as he does, he is only expressing his need, 
when he imagines his stationary system S moving under the 
name of S', to incurvate the simultaneity among events. It is 
always the same simultaneity; it would appear such to an 
observer inside S'. But, expressed perspectively from point N, 
it must be bent back in the form of succession. 

Hence, there is really no need to reassure us, to tell us that 
the observer at N' can unquestionably retain part of the fu- 
ture of place P' within his present, but that he can neither 
grasp it nor give any idea of it, and that, consequently, this 
future is as if nonexistent for him. We are quite undisturbed; 
we cannot stuff and reanimate our observer at 2V' drained of 
his content, remake him into a conscious being, a physicist at 
that, without the event of place P', which we just shelved in 
the future, again becoming the present of this place. Basically, 
it is himself whom the physicist at N needs to reassure at this 
point, and it is himself whom he reassures. He has to prove 
to himself that in numbering the event of point P' as he does, 
m locating it in the future of this point and in the present of 
the observer at N', he is not only satisfying the requirements 
of science, but also remaining fully in accord with ordinary 
experience. And he has no trouble in proving this to himself, 
because when he represents everything according to the rules 
of perspective that he has adopted, what is coherent in reality 
continues to be so in the mental view. The same reason that 
eads him to believe that there is ne speed greater than that of 
light, that the speed of light is the same for every observer, 



etc., obliges him to shelve in the future of place F an event 
that is part of the present of the observer at N', which is, more- 
over, a part of his own N observer's present, and which be- 
longs to the present of place P. Strictly speaking, he ought to 
express himself as follows, "I locate the event in the future 
of place F, but since I leave it within the interval of future 

time -, since I do not push it further back, I shall never have 

to imagine the person at AT' as able to perceive what will occur 
at P' and to inform its inhabitants of it." But the way he sees 
things makes him say, "In vain does the observer at N' possess 
something of the future of place P' in his present; he cannot 
study it, influence it, or use it in any way." Certainly, no 
physical or mathematical error will result from this statement; 
but great would be the delusion of the philosopher who 
would take the physicist at his word. 

For the observer at N', therefore, there is not, at M' and 
p > next to events that we consent to leave in the "absolute 
Past" or in the "absolute future," a whole mass of events 
which, past and future at those two points, enter his present 
whenever we attribute the appropriate speed to system S'. 
There is, at each of these points, only one event making up 
a part of the real present of the observer at N', whatever the 
s Peed of the system; it is the very one that, at M and P, is part 
of the present of the observer at N. But this event will be 
noted down by the physicist as located more or less back in 
*e past of M', more or less forward in the future of F, ac- 
cording to the speed attributed to the system. It is always, at 
M ' a nd F, the same couple of events that form together with 
a certain event at N' the present of Paul located at this latter 
point. But this simultaneity of three events appears incurvated 
mt o Past-present-future when beheld in the mirror of mo- 
tlQ n by Peter picturing Paul. 

However, the illusion involved in the current interpretation 
18 so difficult to unmask that it will not be without profit to 
*" a * it from still another direction. Let us imagine anew 
mat system S', identical with system S, has just broken away 


from it and instantly attained its speed. Peter and Paul have 
been merged at point N; here they are, at the same instant, 
separate at N and N', which still coincide. Let us now imagine 
that Peter, in his system S, has the gift of instantaneous vision 
at a distance. If the motion imparted to system S' really ren- 
dered an event in the future of place P' simultaneous with 
what is occurring at N' (and, consequently, with what is oc- 
curring at N, since the dissociation of the two systems takes 
place at the same instant), Peter would witness a future event 
of place P, an event that will not, as before, enter the present 
of the aforesaid Peter; in short, through the intermediary of 
system S', he would read the future of his own system S, not 
certainly for point N where he is, but for a distant point P. 
And the greater the abruptly attained speed of S' the farther 
will his gaze bore into the future of point P. Had he the means 
for instantaneous communication, he would announce to an 
inhabitant of place P what was going to happen at that point, 
having seen it at P>. But hold on! What he perceives at ?', 
m the future of place P', is exactly what he perceives at P, in 
the present of place P. The greater the speed of system S', the 
f^her back in the future of place P' is what he perceives at 
P , but it is ever and anon the same present of point P. Vision 
at a distance, and into the future, does not therefore inform 
him of anything. There is no room for anything in "the inter- 
val of time" between the present of place P and the future, 
identical with this present, of the corresponding place F; 
everything happens as if the interval were nothing. And it 
is, in tact, nothing; it has been expanded out of nothing. But 
it takes on the appearance of an interval through a phenome- 
non of mental optics, analogous to that which separates an 
object from itself, as it were, when a pressure on the eyeball 
makes us see it double. More precisely, the view of system S' 
ZTrT entert *ined is nothing other than that of 

ystem S skewed" in time. This "skewed vision" makes the 
line of simultaneity passing through points M, N, P in system 

lrl»ZV m T ° bHqUe in s y stem 5 '> duplicate of S, the 
greater the speed of system y. the duplica| £ of what is c c- 



curring at M thus finds itself pushed back into the past, the 
duplicate of what is occurring at P, pulled forward into the 
future; but the long and short of it is that we have here only 
an effect of mental torsion. Now, what we say of system S', 
duplicate of S, is true of any other system having the same 
speed; for, once more, the temporal relations of events in S' 
are affected, following the theory of relativity, by the system's 
speed, and by its speed only. Let us then imagine that S' is 
any system and no longer the double of S. If we want to find 
the exact meaning of the theory of relativity, we must first 
have S' at rest together with S without merging with it, then 
have it move. We shall find that what was simultaneity at 
rest remains simultaneity in motion, but that this simultaneity, 
perceived from system S, has simply been skewed; the line 
of simultaneity between the three points M', N', F appears 
turned about N' by a certain angle, so that one of its ex- 
tremities lags behind in the past while the other encroaches 
upon the future. 

We have dwelled upon the "slowing of time" and the 
"breakup of simultaneity." There remains the "longitudinal 
contraction." We shall presently show how it is but the spatial 
manifestation of this double temporal effect. But we can say 
something about it even now. Let there be (Figure 6) two 


Figure 6 


P° in ts A' and B' in the moving system S' which, during its 
Journey, happen to settle over two points A and B in the mo- 
onless system S, of which S' is the duplicate. When these two 
c< »ncidin g s take place, the clocks at A' and B', synchronized, 
of course, by observers attached to S', show the same time, me 



observer, attached to 5 who believes that, in such a case, the 
clock at B' lags behind the one at A', will conclude that B' 
coincided with B only after the moment of the coinciding of 
A' with A, and that, as a consequence, A'B' is shorter than AB. 
Actually, he "knows" this only in the following sense. In order 
to conform to the rules of perspective, which we stated earlier, 
he had to attribute a delay to the coinciding of B' with B over 
the coinciding of A' with A, precisely because the clocks at A' 
and B' showed the same time for the two coincidings. Conse- 
quently, on pain of contradiction, he has to mark off a shorter 
length for A'B' than for AB. Moreover, the observer in S' will 
argue symmetrically. His system is motionless for him; and, 
consequently, S moves for him in an opposite direction from 
the one S' just followed. The clock at A therefore appears to 
him to be lagging behind the clock at B. And, as a result, the 
coinciding of A with A' will have been effected, according to 
him, only after that of B with B', if clocks A and B showed 

me same time at the two coincidings. From which it follows 
that AB must be shorter ^ ^ ^ w ^ ^ ^ 

ea y the same length, or have they not? Let us repeat once 

ZZ^lr ^ hCre CaUin § real what is perceived or per- 
ceptible. We must therefore turn to the observer in S and 5', 
reter and Pa ^ and compare ^ ^ tions 0 f 

Ae two lengths. Now, each of them, when he sees instead of 

iWh i I 5 ' 611 ' WhCn he is referrin S ™d not referred to, 
^mobilizes his system. Each of them assumes that the length 

recinrn, T U " ^ B ° th s y stems < » an actual state of 

cl77t rTl heing interch angeable, since S' is a dupli- 
bv hvnn/h °b S erver's vision of AB is therefore identical, 
Se eau a vT\ W u h thC * ° bserver ' s ™on of A'B'. How can 
Ire X or I° the r° kngths AB and A ' B ' be a«er ted any 
lute m Z 2 I and / bsoluteI y? Equality takes on an abso- 

we declare Z , ° terms c °mpared are identical; and 
Tble Hen J > T* 1 When We assume ih ™ interchange- 
can no m2 T ^ ° f S P edal rdativ "y' the extend6d 
can no more really contract than time slow down or simul- 



taneity actually break up. But, when a system of reference has 
been adopted and thereby immobilized, everything happening 
in other systems must be expressed perspectively, according 
to the greater or lesser difference that exists, on a size-scale, 
between the speed of the system referred to and the speed, 
zero by hypothesis, of the referrer system. Let us not lose sight 
of this distinction. If we have a living John and James step 
out of the painting where the one occupies the foreground 
and the other the background, let us be careful not to leave 
James a midget. Let us give him, like John, his normal size. 

To sum it all up, we have only to return to our initial hy- 
pothesis of the physicist attached to the earth, repeatedly per- 
forming the Michelson-Morley experiment. But we shall now 
imagine him preoccupied above all with what we are calling 
real, that is, with what he perceives or can perceive. He re- 
mains the physicist, not losing sight of the need to obtain a 
coherent mathematical representation of the whole. But he 
wants to help the philosopher in his task; and his gaze never 
leaves the moving line of demarcation that separates the sym- 
bolic from the real, the conceived from the perceived. He will 
then speak of "reality" and "appearance," of "true measure- 
ments" and "false measurements." In short, he will not adopt 
toe language of relativity. But he will accept its theory. The 
translation of the new idea into the old language with which 
he will furnish us will make clearer what we can keep and 
what we ought to change of what we had previously accepted. 

Accordingly, revolving his apparatus 90°, at no time of the 
year does he observe any shift in the interference bands. The 
speed of light is thus the same in every direction, the same for 
ev ery speed of the earth. How explain this fact? 

"The fact is fully explained," our physicist will declare. 
There is no difficulty, a problem is raised only because we 
s P e: * of an earth in motion. But in motion with respect to 
what? Where is the fixed point that it approaches and moves 
J w ay from? This point can have been only arbitrarily chosen. 
1 a m then free to decree that the earth shall be this point, and 


to refer it to itself, as it were. There it is, motionless, and the 
problem disappears. 

Nevertheless, I have one misgiving. How embarrassing if the 
concept of absolute immobility did take on meaning all the 
same, a definitively fixed landmark having somewhere come to 
light? Without even going that far, I have only to look at the 
stars to see bodies moving with respect to the earth. The physi- 
cist attached to one of these extraterrestrial systems, reasoning 
as I do, will consider himself motionless in turn and rightly 
so; he will then make the same demands of me as would the 
inhabitants of an absolutely motionless system. He will tell 
me, as they would have, that I am deceiving myself, that I have 
no right to explain the equal speed of propagation of light in 
every direction by my immobility, for I am in motion. 

But here then is how I reassure myself. No extraterrestrial 
onlooker will ever reproach me, ever catch me in error, be- 
cause, examining my units of measurement for space and time, 
observing the moving of my instruments and the rate of my 
clocks, he will note the following: (1) I undoubtedly attribute 
the same speed to light as he does, even though I am moving 
in the direction of the beam of light and he is motionless; 
but this is because my units of time then appear to him longer 
than his own; (2) I believe I have established that light is 
propagated with the same speed in every direction; but this is 
because I am measuring distances with a ruler whose length he 
sees changing with its orientation; (3) do I always find that 
ignt has the same speed, even if I happen to measure it be- 
tween two points of its journey on the earth by noting on 
clocks respectively located at these two places the time it takes 
to traverse the interval? but this is because my two clocks have 
been synchronized under the assumption that the earth was 
motionless. As it is in motion, one of the clocks happens to lag 
behind the other with every increase in the earth's speed. This 
2 ? T ^ me £o think th *t the time taken by 
ever ,1° f aVCrSe ^ intCrVal is one that corresponds to an 
ever constant speed. Hence, I am covered. My critic will find 
my conclusions sound although, from his point of view, which 


is now alone legitimate, my premises have become false. At 
most, he will reproach me for believing that I have actually 
established the constancy of the speed of light in every direc- 
tion; according to him, I assert this constancy only because my 
mistakes in measuring time and space so compensate each 
other as to give a result like his. Naturally, in the representa- 
tion of the universe that he will build up, he will have my time 
and space lengths appear as he has just recorded them and 
not as I had recorded them myself. I shall have been judged 
to have mistaken my measurements throughout. But no matter, 
since my result is admitted to be correct. Besides, if the ob- 
server merely imagined by me became real, he would find him- 
self confronted by the same difficulty, would have the same 
misgivings, and would reassure himself in the same way. He 
would say that, moving or motionless, measuring truly or 
falsely, he gets the same physics as I do and ends up with uni- 
versal laws." 

In still other terms: given an experiment such as that of 
Michelson and Morley, things happen as if the theoretician of 
relativity were pressing one of the experimenter's eyeballs and 
thus causing a special kind of diplopia; the image first per- 
ceived, the experiment first begun, doubles into a phantasmal 
image where duration slows down, where simultaneity incur- 
ves into succession, and where, for that very reason, lengths 
cha nge. This diplopia, artificially induced in the experimenter, 
ls to reassure him, or rather, to secure him against the risk he 
thinks he is running (which he really would be running in 
ce "ain cases) in arbitrarily making himself the center of the 
world, in referring everything to his personal system of refer- 
ee, and in nevertheless building up a physics that he would 
j*e to be universally valid. He can rest easy from now on; he 
knows that the laws he formulates will be confirmed, no mat- 
ter from what vantage point we view nature. For the phan- 
tasmal image of his experiment, an image which shows him 
h °w this experiment would look, if the experimental device 
w «e in motion, to a motionless observer provided with a new 
system of reference, is no doubt a temporal and spatial distor- 


tion of the first image, but a distortion that leaves the relations 
among the parts of the framework intact, keeps its connections 
just as they are, and lets the experiment go on confirming the 
same law, these connections and relations being precisely what 
we call the laws of nature. 

But our terrestrial observer must never lose sight that, in all 
this, he alone is real, and the other observer, phantasmal. He 
may, moreover, evoke as many of these phantasms as he likes, 
as many as there are speeds, an infinity of them. All will ap- 
pear to him as building up their representation of the uni- 
verse, changing the measurements he has taken on earth, 
obtaining for that very reason a physics identical with his. 
From then on, he will work away at his physics while remain- 
ing unreservedly in his chosen observation post, the earth, and 
will pay them no more heed. 

It was nonetheless necessary that these phantasmal physicists 
be evoked; and the theory of relativity, by furnishing the real 
physicist the means for finding himself in agreement with 
them, has caused science to take a great step forward. 

We have just located ourselves on the earth. But we could 
just as easily have chosen any other point in the universe. At 
each of these there is a real physicist drawing a host of phan- 
tasmal physicists in his wake, as many as the speeds he im- 
agines. Do we wish, then, to sort out the real? Do we want to 
know whether there is a single time or multiple times? We 
must pay no attention to phantasmal physicists, we must take 
account only of real physicists. We shall ask ourselves whether 
or not they perceive the same time. Now, it is in general diffi- 
cult for the philosopher to declare with certainty that two 
people hve the same rhythm of duration. He cannot even give 
this statement a rigorous, precise meaning. Yet he can do so in 
the hypothesis of relativity. Here the statement takes on a very 
clear meaning and becomes certain when we compare two sys- 
tems m a state of reciprocal and uniform motion; the observers 
are interchangeable. That, indeed, is completely clear and cer- 
tain only in the hypothesis of relativity. Anywhere else, two 
systems, however similar, usually differ in some way, since they 


do not occupy the same place with respect to the privileged 
system. But the doing away with the privileged system is the 
very core of the theory of relativity. Hence, this theory, far 
from ruling out the hypothesis of a single time, calls for it and 
gives it a greater intelligibility. 


The Light-Figures 

"Light-lines" and rigid-lines-the "light-figure" and the 
space-figure; how they coincide and dissociate; triple 
ettect of the d 1S sociation; (1) transverse effect or "ex- 
pansion of time," (2) longitudinal effect or "breakup 
of simultaneity," (3) transverse-longitudinal effect or 
U>rentz contraction"; true nature of Einstein's time; 
transition to the theory of space-time 

This way of looking at things will allow us to penetrate fur- 
ther into the theory of relativity. We have just shown how 

ctminn f^ lan ° f rdativky CVokes ' in add ^on to his per- 
all i Z °k -° Wn SyStem ' 311 the mental views ascribable to 
dos S ^1 P hyS1 " St ! r P erCCiving that svstem in motion at every 
ZTaf^i mCntaI Views ™y> ^t the different 

Teach 1 Cm ^ S ° Elated as to maintain, inside 

the s^eT T rdati0nS am ° ng them and thus to manifest 

InTvi wT LeT « T ^ *** " *~ ***** 
the inrr^c ? de monstrate, in more concrete fashion, 

2Lsssr f of * e surface - d the u - 

adiudeed C x US mner relations ™ the speed is 

2 genesis o^e ^ ^ thU$ Catch ' as if ° n 

We fhalT ee r P ^ ° f Umes in the theory of relativity. 

™; e : re e r e g ^ ^ - r 

which this theory impL " aMncate Cmain P° SW,ateS 
Here then i s the Michelson-Morley experiment (Figure 7) 



^ a motionless system S. Let us give the name, "rigid-line," 
Let 11 Sh ° n ' t0 a mathematica l line such as OA or OB. 

^ the k ^ ° f Hght that ° VCr k " H g ht - line " 

fitted t° insidC the System ' the two beams ' both 

return ex l fr ° ra ° t0 B and 0 to A ' res P ectivel Y' 

off ers him^h . Up ° n ^enwehres. The experiment therefore 
0 and B a d ° f & d ° Uble H g ht - line stretched between 
equal ^ ° and A > these two double light-lines being 

VoTt^lf endicular to each other - 

s Peedv Wh '"^ ^ s ^ stem at rest > imagine it moving at 

As long V- WU1 bC ° Ur double mental view of it? 
formed eithe "h* " ^ WC Ca " consider indifferently, as 

t *° d °ubi e r 5 y i- w ° singIe ri § id " lines at ri s ht an S les or b Y 

^rigid-figure nCS ' again at right an S les; the light-figure 
^ tWo figure j 01 " 0 ^ As soon as we imagine it in motion, 
•*° lines at S dlSSOciate - The rigid-figure stays composed of 
The d m 3ngleS " But the n g ht - fi g ure becomes dis- 
broken light l° U n llght " Hne stretched along OB becomes a 
4n & 04 b Pr i ne The double light-line stretched 

^^nereauvr* 6 Iight ' Une < the P ortion 

y "es on O'A' but, for greater clarity, we are 


detaching it in the figure). So much for its shape. Let us con- 
sider its size. 

Anyone who would have reasoned a priori, before the 
Michelson-Morley experiment had actually been performed, 
would have said: "I must assume that the rigid-figure remains 
as it is, not only in the two lines remaining at right angles to 
each other but also in their being always equal. That follows 
from the very concept of rigidity. As for the two double light- 
lines, originally equal, I picture them becoming unequal when 
dissociating, as the result of the motion that my thought im- 
parts to the system. That follows from the very equality of 
the two rigid lines." In short, in this a priori argument, based 
upon the old ideas, we would have said: "It is the rigid space- 
figure that imposes its conditions upon the light-figure." 

The theory of relativity, as it has emerged from the ac- 
tually performed Michelson-Morley experiment, consists of 
reversing this proposition and saying, "It is the light-figure 
that imposes its conditions upon the rigid-figure." In other 
words, the rigid-figure is not reality itself but only a mental 
construct; and for this construct it is the light-figure, the 
sole datum, which must supply the rules. 

The Michelson-Morley experiment apprises us, in effect, 
that the two lines O^O',, O x A x O\ remain equal, no matter 
what speed is attributed to the system. It is therefore the 
equality of the two double light-lines that will always be con- 
sidered preserved and not that of the two rigid lines; it is for 
the latter to arrange themselves accordingly. Let us see how 
they do this. To that end, let us closely examine the distor- 
tion of our hght-figure. But let us not forget that everything 
is happening in our imagination, or, rather, in our under- 
standing. In point of fact, the Michelson-Morley experiment 
has been performed by a physicist in his system, and, there- 
ore, in a motionless system. The system is in motion only if 
the physiast mentally leaves it. If he remains there in thought, 

mLTT/ 7 iU ap P^ to his °™ "ytem, but to the 
Michelson-Morley experiment undertaken in another system, 
or, rather, to the image he forms, which he must form, of this 


experiment started elsewhere; for, where the experiment is 
actually performed, it is as yet done by a physicist within the 
system, and, therefore, in a still motionless one. The result is 
that, in all this, it is only a question of adopting a certain 
notation for the experiment we do not perform, in order to 
co-ordinate it with the one we do perform. We are thus simply 
saying that we are not performing it. Never losing sight of 
this point, let us follow the change in our light-figure. We 
shall separately examine the three distortional effects pro- 
duced by motion: (1) the transverse effect, which corresponds, 
as we shall see, to what the theory of relativity calls a length- 
ening of time; (2) the longitudinal effect, which, for it, is a 
breaking up of simultaneity; (3) the twofold transverse-longi- 
tudinal effect, which is "the Lorentz contraction." 


Let us give speed v increasing rates from zero up. Let us 
& ain ourselves mentally to turn out of the original light-figure 
°AB a series of figures in which the divergence between light- 
»nes that first coincided becomes ever more marked. Let us 
also practice making all those which have thus come out of 
11 retreat within the original figure. In other words, let us 
proceed as with a spyglass whose tubes we pull out and then 
telescope. Or better, let us think of that child's toy made of 
Jointed sticks lined with wooden soldiers. When we spread 
*e sticks apart by pulling on the two end ones, they cross 
' lke X's and the soldiers break ranks; when we push them 

ack > al l the sticks come together and the soldiers close ranks. 
L« us clearly repeat that the number of our light-figures is 
infinite and that they are nevertheless but one; their multi- 
ply merely expresses the possible visions had of them by 
Servers to whom they seem to be traveling at different 
JP e eds, that is, the visions that observers moving relative to 

em ha ve; and all these virtual visions telescope, so to speak, 
lnt0 the real vision of the original figure AOB. What con- 
J. Usi0n forces itself upon us regarding the transverse light- 
lme 0 i^O' lf the one which has sprung from OB and could 



return to it, which actually does return to it and becomes one 
with OB the very instant we picture it there? This line is 

equal to — — when the original double light-line was 21. 

Its lengthening therefore represents exactly the lengthening 
of time as given in the theory of relativity. We see from this 
that the theory proceeds as if we were taking the double jour- 
ney of a light beam's departure and return between two fixed 
points as the standard of time. But we then perceive at once, 
intuitively, the relation of multiple times to the single, real 
time. Not only do the multiple times conjured up by the the- 
ory of relativity not disrupt the unity of a real time but they 
even imply and uphold it. The real observer inside his system 
is indeed aware of both the difference between, and the iden- 
tity of, these two different times. He lives a psychological time, 
and, with this time, all the more or less expanded mathemati- 
cal times merge; for in proportion as he spreads apart the 
hinged sticks of his toy-in the measure that he mentally 
accelerates the motion of his system-the light-lines lengthen, 
but they all fill the same lived duration. Without this unique, 
lived duration, without this real time common to all the math- 
ematical times, what would it mean to say that they are con- 
temporaneous, that they abide within the same interval? What 
meaning could we really find in such a statement? 

Let us suppose (we shall return to this point shortly) that 
uk ; observer m 5 is accustomed to measuring his time by a 
igm-lme xn other words, to pasting his psychological time 
to .his hght-hne OB. Necessarily, psychological time and light- 
nrV COn wf in thC motion1 ^ system) will be synonymous 
tor him. When, imagining his system in motion, he will think 

bur h P § n f 35 l0nger ' hG WiU sa y tha < «™ has lengthened; 
is a tZZ 1 1 that k is no Ion S er Psychological time. It 

mathZ, , t "° IOngCr ' 38 before ' b °* psychological and 
pableTf h 1 beC ° me CXclusiveI y mathematical, inca- 
sdousni g ,5 ny ° ne ' S P^ogical time. As soon as a con- 
saousness would wish to live one of these lengthened times 
i x. "2*2, etc., these latter would immediately retract into 



OB, since the light-line would then no longer be perceived in 
imagination but in reality, and the system, until then only 
mentally set in motion, would claim actual immobility. 

In short, therefore, the thesis of relativity here clearly inti- 
mates that an observer inside system S, picturing this system 
in motion at every possible speed, sees the mathematical time 
of his system lengthening with an increase in speed if this sys- 
tem's time had been identified with the light-lines OB, O-fi^, 
0 2 B 2 , etc. All these different mathematical times are contempo- 
raneous, in that all abide within the same psychological dura- 
tion-that of the observer in S. They are only fictional times, 
moreover, since they cannot be lived differently from the first 
by anyone, neither by the S observer who perceives them all 
within the same duration, nor by any other real or possible 
observer. They hold on to the name "time" only because the 
first of the series, namely OB, measured the psychological 
duration of the observer in S. Then, by extension, we still 
apply the term "time" to the now lengthened light-lines of the 
supposedly moving system, forcing ourselves to forget that they 
all abide within the same duration. Let us, by all means, keep 
*e name "time" for them: they are conventional times by defi- 
nition, since they measure no real or possible duration. 

fi ut how explain, in a general way, this rapprochement 
bet ween time and light-line? Why has the first of the light- 
lines, OB, been pasted by the observer in S to his psychological 
dur ation, imparting then the name and appearance of time to 
the successive lines O^, 0 2 B 2 , etc., by a kind of contamina- 
tion? We have already answered this question implicitly; it 
wil1 nevertheless not be without profit to submit it to a new 
lamination. But let us first see-while continuing to make a 
'■ght-line of time-the second effect of the distortion of the 


As the light-lines that coincided in the original figure grow 
^Aer apart( the inequality becomes accentuated between 
Wo kngitudinal light-lines, such as 0 1 A 1 and A x O v ong 



nally merged with the double light-line OA. Since, for us, 
the light-line is always time, we shall say that the moment A x 
is no longer in the middle of time interval O x A^O\, when 
the moment A was in the middle of the OAO interval. 
Now, whether the observer in system S assumes his system 
to be at rest or in motion, his assumption, a mere mental 
act, in no way influences his system's clocks. But it does in- 
fluence their agreement, as we see. The clocks do not change; 
time changes. It is distorted and breaks up among them. 
It was equal times which, so to speak, went from O to A 
and returned from A to O in the original figure. Now the 
departure takes longer than the return. We easily see, more- 
over, that the second clock will lag behind the first by either 
1 lv lv j 8 1 

1 W c 2 ° r ' de P endi ng upon whether we record it in 

seconds of the motionless system or the moving system. Since 
the clocks stay as they were, run as they have, preserve, conse- 
quently, the same relations with one another and remain 
synchronized as originally, they are found, in the mind of our 
observer, to lag more and more behind one another in propor- 
tion as his imagination accelerates the system's motion. Does 
he perceive himself motionless? There actually is simultaneity 
between the two instants when the clocks at O and A show the 
same time. Does he imagine himself in motion? These two 
instants, underscored by the two clocks showing the same time, 
cease by definition to be simultaneous, since the two light-lines 
have changed from equal to unequal. I mean that it was first 
equality, and now inequality, which has just slipped between 
the two clocks, they themselves not having budged. But have 
this equality and inequality the same degree of reality if they 
claim to apply to time? The first was at one and the same time 
an equality of light-lines and psychological durations, that is, 
oi time in everyone's sense of the word. The second is nothing 
more than an inequality of light-lines, that is, of conventional 
times; it arises, however, among the same psychological dura- 
tions as the first. And it is just because psychological duration 



continues to exist, unchanged, throughout all the successive 
imaginings of the observer, that he can consider all his im- 
agined, conventional times as equivalent. He stands before 
figure BOA; he perceives a certain psychological duration that 
he measures by the double light-lines OB and OA. Now, with- 
out ceasing to look, therefore always perceiving this same 
duration, he sees, in his imagination, the double light-lines 
dissociate as they lengthen, the double longitudinal light-line 
splitting into two lines of unequal length, the inequality in- 
creasing with the speed. All these inequalities have come out 
of the original equality like the tubes out of a field glass; if it 
suits him, they will all instantly re-enter by telescoping. They 
are equivalent for him precisely because the true reality is the 
original equality, that is, the simultaneity of the moments indi- 
cated by the two clocks, and not the succession, purely imagi- 
nary and conventional, which the merely imagined motion of 
the system and the resultant breakup of its light-lines en- 
gender. All these breakups and successions are hence virtual; 
only the simultaneity is real. And it is because all these virtu- 
al ities, all these varieties of dislocation abide inside the really 
Perceived simultaneity that they are mathematically substi- 
tatable for it. All the same, there are, on the one hand, the 
lr nagined, the merely possible, while, on the other hand, are 
the perceived and the real. 

, Now, the fact that, consciously or not, the theory of rela- 
tivity substitutes light-lines for time places one of its principles 
ln ful1 view. In a series of studies on the theory of relativity, 1 
Edouard Guillaume has maintained that it essentially consists 
°f making a clock out of the propagation of light, instead of 
*e rotation of the earth. We believe there is much more than 
tha t in the theory of relativity. But we believe there is at least 
Jnat. And we shall add that, in isolating this ingredient, one 
but emphasizes the theory's importance. In fact, still on this 
P 0l nt, one thus establishes that the theory is the natural and 
Perhaps necessary outcome of a long development. Let us 
^ Rune de mdtaphysique (May-June 1918, and October-December 1920). 
a ™one de la Relativiti (Lausanne, 1921). 



briefly recall the penetrating and profound thoughts that 
Edouard le Roy set forth not long ago on the gradual perfect- 
ing of our means of measurement, especially the measurement 
of time. 2 He showed how a certain method of measuring 
enables us to establish laws and how these, once laid down, 
can react upon the method of measurement and compel it to 
be modified. With more particular reference to time, we have 
used the sidereal clock in the development of physics and 
astronomy; specifically, we have discovered the Newtonian law 
of attraction and the principle of the conservation of energy. 
But these results are incompatible with the constancy of the 
sidereal day, because, according to them, the tides must act as 
a brake upon the earth's rotation. Thus, the use of the sidereal 
clock leads to consequences which require the adoption of a 
new clock. 3 There is no doubt but that the progress of physics 
tends to present us with the optical clock-meaning the propa- 
gation of light-as the ultimate clock, the one that is the term 
of all those successive approximations. The theory of relativity 
records this outcome. And, as it is of the essence of physics to 
identify the thing with its measurement, the "light-line" be- 
comes both the means of measuring time and time itself. But 
then, since the light-line elongates, while remaining itself, 
when we imagine as in motion yet leave at rest the system in 
which it is observed, we shall obtain multiple, equivalent 
times; and the hypothesis of the plurality of times, character- 
istic of the theory of relativity, will appear as conditioning the 
general evolution of physics as well. Times thus defined will 
indeed be physical times. 4 They will be only conceived times, 

2 BMetin de la Society francaise de philosophic February 1905. 

«CL Ermle Borel, L'espace et le temps (Paris: F. Mean, 1922) p. 25. 
We have called them "mathematical," in the course of the present 
essay, m order to avoid any confusion. We are, indeed, continually com- 
paring them with psychological time, distinguishing between the mathe- 
matical and the psychological and keeping this distinction ever in mind. 
Now, the difference between the psychological and the mathematical is 

tM 1 ''."^ m '! Ch kSS S ° b6tWeen ^ Psychological and the physical. The 
term physical time" might at times have had a double meaning; "mathe- 
matical ume can have nothi ambi ^ , 


however, all except one, which will actually be perceived. The 
latter, always the same, is the time of common sense. 

Let us sum up briefly. For a common-sense time, which can 
always be converted into psychological duration and which 
thus happens to be real by definition, the theory of relativity 
substitutes a time that can be converted into psychological 
duration only in the case of the system's immobility. In all 
other cases, this time, which was both light-line and duration, 
is no more than light-line-an elastic line that stretches as the 
speed attributed to the system increases. It cannot correspond 
to a new psychological duration, since it continues to fill this 
same duration. But small matter; the theory of relativity is a 
physical theory; it tends to ignore all psychological duration, 
as much in the first case as in all the others, and to retain of 
time nothing more than the light-line. As the latter either 
lengthens or contracts with the speed of the system, we thus 
obtain multiple, contemporaneous times. And that seems para- 
doxical because real duration continues to haunt us. But, on 
Ae other hand, it becomes very simple and quite n * tur! j| 
when we substitute an extensible light-line for time and call 
simultaneity and succession instances of equality and inequal- 
ity between light-lines whose interrelations evidently change 
with the system's state of rest or motion. 

But these reflections upon light-lines would be incomplete 
lf we limited ourselves to studying the transverse and longi- 
tudinal effects separately. We must now be present at their 
compounding. We shall see how the connection that must 
always obtain between longitudinal and transverse light-lines, 
whatever the system's speed, entails certain consequences re- 
garding rigidity, and, therefore, extension as well. We snail 
thus obtain a lifelike picture of the interweaving of space an 
li «ie in the theory of relativity. This interweaving appears 
cl early only after we have reduced time to a light-line. »y 
"Kans of the light-line, which is time but remains »» btm « 
W "Pace, which lengthens as a result of the system s mo ion 
a »d thus gathers up, on the way, the space with which it makes 
tl *e, we shall grasp, in concreto, in everyone's time and space, 


the very simple, initial fact expressed by the conception of a 
four-dimensional space-time in the theory of relativity. 


"LORENTZ contraction" 

The special theory of relativity, we said, consists, in essence, 
of first picturing the double light-line BOA, then distorting it 
into such figures as 0 1 B 1 A 1 0\ through the system's motion, 
finally in making all these figures return, pull out, and return 
again one inside the other, while accustoming ourselves to 
thinking that they are both the first figure and the figures 
pulled out of it. In short, after mentally imparting every pos- 
sible speed to the system, we entertain every possible vision of 
one and the same thing, this thing being deemed to coincide 
with all these visions at one and the same time. But the thing 
with which we are thus dealing is essentially a light-line. Let 
us consider the three points O, B, A of our first figure. Ordi- 
narily, when we call them fixed points, we deal with them as 
if they were connected by rigid bars. In the theory of relativity, 
the bond becomes a ribbon of light which we would emit from 
O to 2? m such a way as to have it return upon itself and be 
caught again at O, another ribbon of light being emitted be- 
tween O and A, touching A only to return to O. This means 
that time will now be amalgamated with space. Under the 
rigid bar" assumption, the three points were connected in the 
instantaneous, or, if you prefer, in the eternal, in a word, out- 
side ot time; their relation in space was unchanging. But here, 
with elastic and distortable shafts of light which are repre- 
sentative of time, or, rather, are time itself, the relation of the 
three points falls under time's dependency. 

To understand clearly the "contraction" that ensues, we 
nave only to examine the successive light-figures, realizing that 
they are figures, tracks of light which we take in at a glance, 
and that we shall nevertheless have to treat the lines in them 
as if they were time. These light-lines alone being given, «e 
must mentally reconstitute the space-lines, which will in gen- 


eral no longer be perceived in the figure itself. They can be 
no more than inferred, mentally reconstructed. The one excep- 
tion, of course, is the light-figure of the system ruled motion- 
less; thus, in our first figure, OB and OA are both flexible 
light-lines and rigid space-lines, the apparatus BOA being 
ruled at rest. But in our second light-figure, how are we to 
picture the apparatus with its two rigid space-lines supporting 
the two mirrors? Let us consider the position of the apparatus 
the moment B reaches B x . If we drop perpendicular B x O" x on 
0 X A X , can we say that figure B x O" x A x is that of the apparatus? 
Clearly not, because if the equality of light-lines 0 1 B 1 and 
O'A shows us that moments 0" x and B x are truly contempo- 
raneous, if 0" X B X really retains its character of a rigid space- 
line, if, therefore, 0" 1 B 1 really represents one of the arms of 
the apparatus, the inequality of light-lines O x A x and^O^i 
shows us, on the other hand, that the two moments 0" x and 
A are successive. The length 0" 1 A 1 therefore represents the 
other arm of the apparatus plus the distance covered by the 
apparatus during the interval of time that separates moment 
°"i from moment A x . Hence, to obtain the length of this sec- 
ond arm, we must take the difference between 0" 1 A 1 and * e 
distance covered. This is easy to calculate. The length 0" X A X 
is the arithmetical mean between O x A x and 0\A X , and as the 

sum of these last two lengths is equal to -j=^' since the 

complete line O x A x O\ represents the same time as line O x B x O\, 
*e see that the length of 0" X A X is -==• As for the space 

covered by the apparatus in the interval of time between mo- 
mei »s 0>\ and A x , we shall estimate it at once by observing 
th at this interval is measured by the slowing of the ciock 
Seated at the extremity of one of the apparatus arms over 

1 W -pjje 

clock located at the other, that is, by -7==i ' c 2 ' 



1 lv 2 

distance covered is therefore ——= • — And, consequently, 

the length of the arm, which was I when at rest, becomes 

I lv 2 i 2 

, , that is, L i _}L. We thus actually redis- 

cover the "Lorentz contraction." 

We see what this contraction means. The identification of 

time with the light-line causes the system's motion to have a 

double effect upon time: expansion of the second, breakup of 

simultaneity. In the difference I lv 2 , „ 

the first term 

corresponds to the expansion effect, the second, to the breakup 
effect. In both cases, we can say that time alone (fictional 
time) is involved. But this combination of effects in time gives 
what we call a contraction of length in space. 

We then grasp the very essence of the theory of relativity. 
It may be expressed in ordinary terms in this way: "Given a 
coinciding, at rest, of the rigid space-figure with the flexible 
light-figure, given, on the other hand, an ideal dissociation of 
these two figures as the result of a motion mentally attributed 
to the system, the successive distortions of the flexible light- 
figure at different speeds are all that count: the rigid space- 
figure will accommodate itself as best it can." As a matter of 
fact, we see that, during the system's motion, the longitudinal 
zigzag of light must keep the same length as the transverse 
zigzag, since the equality of these two times comes before all 
else. As, under these circumstances, the two rigid space-lines, 
the longitudinal and the transverse, cannot themselves remain 
equal, it is space that must give way. It will necessarily give 
way, the rigid diagram in lines of pure space being deemed 
only the registering of the global effect produced by the vari- 
ous changes in the flexible figure, that is, by the light-lines. 


Four-Dimensional Space-Time 

How the idea of a fourth dimension is ushered in; how 
immobility is expressed in terms of motion; how time 
amalgamates with space— the general conception of a 
four-dimensional space-time; what it adds to and sub- 
tracts from reality; twofold illusion to which it exposes 
us; the special character of this conception in the theory 
of relativity; particular error that we risk committing at 
this point; the real and the virtual; what the space-time 
amalgam actually represents 

Let us now take leave of our light-figure with its successive 
distortions. We had to use it to give body to the abstractions 
of the theory of relativity and to bring out the postulates it 
implies. The relation previously established by us between 
multiple times and psychological time has perhaps become the 
dearer for it. And perhaps we have seen the door half opening 
through which the idea of a four-dimensional space-time will 
be introduced into the theory. It is to space-time that we shall 
now turn our attention. 

The analysis just completed has already shown how this 
theory treats the relation of the thing to its expression. Ine 
^Ing is what is perceived; the expression is what the mind 
Puts in place of the thing to make it amenable to calculation. 
The thing is given in a real vision; the expression corresponds 
at most to what we call a "phantasmal vision." Ordinarily, we 
c °ncei ve of phantasmal visions as ephemeral, surrounding me 
st able and firm nucleus of real vision. But the essence ot tne 
* e °ry of relativity is to accord all these visions equal rann 
The vision we call real would be only one of the phantasmal 



visions. This is all right in the sense that there is no way 
mathematically to express the difference between the two. But 
we must not conclude from that to a likeness in kind. Yet this 
is what we do when we confer a metaphysical meaning upon 
Minkowski's and Einstein's four-dimensional space-time con- 
tinuum. Let us indeed see how this notion of space-time arises. 

To that end, we have only to determine with precision the 
nature of the "phantasmal visions," in the case in which an 
observer inside a system S', having really perceived an invari- 
able length Z, would conceive the invariability of this length 
while mentally locating himself outside the system and then 
imagining it endowed with every possible speed. He would 
say to himself: "Since a line A'B' in the moving system S', when 
passing before me in the motionless system in which I install 
myself, coincides with a length I of this system, it is because that 

line, at rest, is equal to 1 I. Let us consider the square 
L * = ^2 " 12 of this magnitude. How much greater is it than 

l ~V 2 

the square of 1? By the quantity 1 . ^ , which can be 

written as c 2 ' 1 lv 

c 2 

But, 1 • ^ is the exact 

' " c 2 

measure of the interval of time T which elapses for me, trans- 
ported into system S, between two events respectively occur- 
ring at A' and B' which would appear simultaneous to me if 
I were in system S'. Hence, as the speed of S' increases from 
zero, the interval of time T broadens between the two events 
occurring at points A' and B', given in S' as simultaneous; but 
things so happen that the difference L 2 - c 2 T 2 remains con- 
stant. It is this difference that I formerly called I 2 ." Thus, 
taking c as the unit of time, we can say that what is given to 


a real observer in S' as the fixity of a spatial magnitude, as the 
invariability of a square P, would appear to an imaginary ob- 
server in S as the constancy of the difference between the 
square of a space and the square of a time. 

But we have just taken a special case. Let us generalize the 
question and first ask ourselves how the distance between two 
points in a physical system S' is expressed with respect to rec- 
tangular axes located in this system. We shall then try to find 
out how it will be expressed with respect to axes in system S 
with respect to which S' would become mobile. 

If our space were two-dimensional, reduced to the size of the 
present page, if the two points considered were A' and B', 
whose respective distances from the axes O'Y' and O'X' are 
x \, y\ and x' 2 , y' 2 , it is clear that we would have 

^B» = (x' 2 -x' 1 ) 2 + (/2-y'i) 2 - 
We could then consider any other system of axes motionless 
with respect to the first and thus give values for x\, x' 2 , y'v y'2 
which would be generally different from the first: the sum of 
Ae two squares (x' 2 -*'i) 2 and (y' 2 -y\) 2 would reraain the 
same, since it would always be equal to WW*. Likewise, in a 
three-dimensional space, points A' and B' being then no longer 
assumed on plane X'O'Y', being now defined by their distances 
y\, * u x' 2) y' 2) z' 2 from the three faces of a trihedral m- 
rectangle whose vertex is O', we would ascertain the invariance 
°f the sum (x' 2 - x\f + (y' 2 - y\f + (z' 2 - z\f. It is by this very 
invariance that the fixity of the distance between A' and B' 
w ould be expressed for an observer located in 5'. 

fi ut let us suppose that our observer mentally enters system 
s with respect to which S' is ruled in motion. Let us also sup- 
pose that he refers points A' and B' to axes located in his new 
system, placing himself, moreover, in the simplified circum- 
stances we described further back when we were working out 
the Lorentz equations. The respective distances from points 
A ' a nd B' to the three rectangular planes intersecting at 6 wu 
»* *i, y lt z i; x 2 , y 2 , z 2 . The square of the distance A B 
between our two points will, moreover, still be given as a sum 
of three squares (x 2 - + (y 2 -ytf + ft-*) 1 - But ' ZCC ° ld ' 



ing to the Lorentz equations, even if the last two squares of 
this sum are identical with the last two of the preceding sum, 
this does not hold for the first, because these equations give us 

for x 2 and x 1( respectively, the values * 

^ _ (x\ + vf) and 

1 (x' 2 + vt'); so that the first square will be — ^( x '2 - *'i) 2 - 

We naturally find ourselves confronting the particular case 
which we were examining just before. We had, in fact, been 
considering a certain length A'B' in system 5', that is, the dis- 
tance separating two instantaneous and simultaneous events 
occurring at A' and B', respectively. But we now wish to gen- 
eralize the question. Let us therefore suppose that the two 
events are successive for the observer in S'. If one occurs at 
moment t\, and the other, at moment t' 2 , the Lorentz equa- 
tions will give us 

*i = ( x 'i + vt\) 


*a = - F L=(x' 2 + t;f' 2 ) 


so that our first square will become 


and our original sum of three squares will be replaced by 

a magnitude that depends upon v and is no longer invariant. 

But if, in this expression, we look at the first term 
-—^[( x '2-x\) + v(t f 2 -t' 1 )]2 i which gives us the va i u e of 



(x 2 -*i) 2 , we see 1 that it exceeds (x' 2 -x'i) 2 b Y the quantity 



\- V - 


Now, the Lorentz equations give: 

We therefore have 

(* 2 - *x) 2 - (x' 2 - X\f = (*(t a - - <*<f 2 - t\f 


(X, - Xl f - C 2(t 2 - tj* = (X' 2 - X\f - C 2 (f 2 - t\Y 

or finally 

x a )» + (y 2 - yi )a + (z 2 - Zl ) 2 - c*(t 2 - ttf 

= (x' a - x'tf + (y' a - y^) 2 + (z' 2 - z^) 2 - c\t\ - t\f 
a result which could be worded as follows: If the observer in S' 
had considered, instead of the sum of three squares 

(*Wi) a + </»-/i) a + (*Wi) > 

*e expression 

, _ (*' S ~X' 1 )2 + (y' a -y' 1 )2 + ( Z ' a _ Z ' 1 )2- C 2(t' a -r i ) a 

w which a fourth square enters, he would have re-established, 
trough the introduction of time, the invariance that had 
ceased to exist in space. 

Our calculations may have appeared a bit clumsy. And so 
toey actually are. Nothing would have been simpler than to 
State at on ce that the expression 

(*2 - x x ) 2 + (y 2 - y,Y + (z 2 - Zl ) 2 - c 2 (<2 - h) 2 
°® not change when we subject its component terms to the 
entz transformation. But that would have been to accord 
^ Ual ran k to every system in which every measurement is 
eemed t0 have been made. The mathematician and the physi- 
^ st must do so, since they are not seeking to interpret the 
Pace-time of the theory of relativity in terms of reality but 
lm Ply to make use of it. On the other hand, our own aim is 
° ne ran ver »fy this easily enough. 



this very interpretation. We therefore had to set out from 
measurements taken in system S' by the observer in S'— the only 
real measurements attributable to a real observer— and to con- 
sider the measurements made in other systems as alterations 
or distortions of the former, alterations and distortions so co- 
ordinated that certain connections among the measurements 
remain the same. The detour we just made was therefore 
necessary to preserve the S' observer's central position and thus 
set the stage for the analysis of space-time, which we shall pre- 
sent shortly. It was also necessary, as we shall see, to establish 
a distinction between the case in which the observer in S' per- 
ceived events A' and B' as simultaneous, and the case in which 
he notes them down as successive. This distinction would have 
vanished if we had made simultaneity only the special case in 
which t' 2 - t\ = 0; we would thus have reabsorbed it into suc- 
cession; every difference in kind would again have been sup- 
pressed between the measurements really made by the observer 
in S' and the merely imagined measurements that observers 
outside the system would make. But small matter for the 
moment. We are merely showing how the theory of relativity 
is actually guided by considerations that precede the positing 
of a four-dimensional space-time. 

We said that the expression of the square of the distance 
between two points A' and B', referred to two axes at right 
angles in a two-dimensional space, is (x 2 - x x ) 2 + (y 2 - yi) 2 > if 
*i> Ju *2> J2 are their respective distances from the two axes. 
We added that in a three-dimensional space this expression 
would become (x 2 - Xl )» + (y 2 - yi y + ( Za _ Zl )2. Nothing prevents 
us from imagining spaces of 4, 5, 6 ... n dimensions. The 
square of the distance between two points would be given in 
them by a sum of 4, 5, 6 ... n squares, each of these squares 
being that of the difference between the distances from points 
A' and B' to one of the 4, 5, 6 ... n planes. Let us then con- 
sider our expression (x 2 - Xl )* + (y 2 _ y j2 + ^ _ Zi)2 _ c ^ h _ tl ) 2 . 
If the sum of the first three terms were constant, it could 
express the constancy of the distance, as we conceived it in our 
three-dimensional space before the theory of relativity. But in 
essence the latter consists in saying that we must introduce the 



fourth term to get this constancy. Why would this fourth term 
not correspond to a fourth dimension? Two considerations 
seem at once to be opposed to this, if we hold to our expression 
for distance: on the one hand, the square (f 2 - *i) 2 is preceded 
by a minus instead of a plus sign; and, on the other, it is 
affected by a coefficient c 2 different from unity. But as, on a 
fourth axis that would be representative of time, times would 
necessarily have to be conveyed as lengths, we can rule that, 
on this axis, a second will have the length c: our coefficient 
will thus become unity. Moreover, if we consider a time r such 
that we have t = i-y^l and if, in a general way, we replace t 
by the imaginary quantity T^/^l, our fourth square will be 
-t 2 , and we shall then really be dealing with a sum of four 
squares. Let us agree to designate by Ax, Ay, Az, At the four 
differences x 2 - x v y 2 - y u z 2 - z u t 2 - r v which are the respec- 
tive increments of x, y, z, t when we pass from x x to x 2 , from 
fi to y 2 , from to z 2 , from t x to t 2 ; and let us designate by As 
interval between the two points A' and B'. We shall have: 
As 2 = Ax 2 + Ay 2 + Az 2 + At 2 . 
And from then on nothing will prevent us from believing 
at s is a distance, or, rather, an interval, in both space and 
time: the fourth square would correspond to the fourth dimen- 
f° n of a s pace-time continuum in which time and space would 
^ amalgamated. 

Nor ls tne *"e anything to keep us from imagining the two 
P° ln ts A' and B' as so infinitely adjacent that A'B' may as well 
^ a curve element. A finite increase like Ax will then become 

n "ifinitesimal increase dx and we shall have the differential 

^ ds2 = dx 2 + dy 2 + dz 2 + dr 2 

itel Cil we can rise again through a summation of infin- 
y small elements, through "integration," to the interval s 

both"* 11 tW ° P ° intS of ' this time ' any line at a11 ' occu Py ing 
space and time, which we shall call AB. We shall write 

■■ f B ^dx 2 + dy 2 + dzHd^, 
J A 



an expression of which we must be cognizant, but to which we 
shall not return in what follows. We shall gain more by mak- 
ing direct use of the considerations that have led us to it. 2 

We have just seen how the notation of a fourth dimension 
is introduced automatically, so to speak, into the theory of 
relativity. This undoubtedly accounts for the oft-expressed 
opinion that we are indebted to this theory for the earliest 
suggestion of a four-dimensional environment merging time 
and space. What has not been sufficiently noted is that a fourth 
dimension of space is suggested by every spatialization of time; 
it has therefore always been implicit in our science and lan- 
guage. Actually, we could sift it out of the usual conception of 
time in a more precise, at least more imagistic, form than out 
of the theory of relativity. But, in the usual conception, the 
comparison of time to a fourth dimension is understood, 
whereas the physics of relativity is obliged to introduce it into 
its calculations. And this leads to the double effect of endosmo- 
sis and exosmosis between time and space, to their reciprocal 
encroachment, which the Lorentz equations appear to express: 
it now becomes necessary, in locating a point, to indicate ex- 
plicitly its position in time as well as in space. Nonetheless, 
Minkowski's and Einstein's space-time remains a species of 
which the ordinary spatialization of time in a four-dimensional 
space is the genus. The course we have to follow is then com- 
pletely laid out. We must begin by seeking the general mean- 
ing of the introduction of a four-dimensional environment 
that would unite time and space. Then we shall ask ourselves 
what we add to, or subtract from, this meaning when we con- 
ceive the relation between spatial and temporal dimensions 
in the manner of Minkowski and Einstein. Even now, one 
begins to see that, if the popular conception of a space joined 
to spatialized time quite naturally takes mental shape as a 

2 The reader who is something of a mathematician will have noticed 
that the expression *• = dx* + dy» + *• - <* dt * can be considered, as it stands, 
as corresponding to a hyperbolic space-time. Minkowski's artifice, described 
above, conswts in giving Euclidean form to this space-time by the substi- 
tution of the imaginary variable ct y^T for ^ variable fc 



four-dimensional environment, and if this environment is 
fictional because it merely symbolizes the convention of 
spatializing time, the same is true for the species of which this 
four-dimensional environment is the genus. In any case, species 
and genus will perforce have the same degree of reality and 
the space-time of the theory of relativity will hardly be any 
more incompatible with our long-standing concept of duration 
than was a four-dimensional space-and-time symbolizing both 
ordinary space and spatialized time. Still, we cannot dispense 
with a more detailed examination of Minkowski's and Ein- 
stein s space-time, when once we have turned our attention to 
a general four-dimensional space-and-time. Let us first apply 
ourselves to the latter. 

We have difficulty in imagining a new dimension if we set 
out from a three-dimensional space, since experience does not 
reveal a fourth. But nothing is simpler if it is a two-dimen- 
sional space that we endow with this added dimension. We can 
wnjure up flat beings, living on a surface, merging with it, 
be" 6 ° f ° nly two dim ensions of space. One of them will have 

ee n led by his calculations to postulate the existence of a 
d di mension. His fellow beings, shallow in the double 
*J*e of the word, will no doubt refuse to heed him; he him- 
hav^ succeed in imagining what his understanding will 
j? Ve b f en abl e to conceive. But we, who live in a three- 
^ensional s P ace > would have the actual perception of what 
^would merely have represented as possible: we would be 
to give an exact account of what he would have added 
producing a new dimension. And, as we ourselves would 
are to^ S ° methin S of the kil "l if we imagined, limited as we 
o three dimensions, that we were immersed in a four- 
we ^ S1 ° nal env ironment, it would be almost in this way that 
unim- picture this fourth dimension that first seemed 
j or a S ina ble. True, this would not be quite the same thing. 
&e mi ° f m ° re than three dimensions is a mere idea in 
. and cannot correspond to any reality. Whereas three- 
ia wh Sl °J al s P ace « Aat of our experience. Therefore, when, 
follows, we use our actually perceived three-dimen- 



sional space to give a body to the formulations of a mathe- 
matician subject to a flat universe— formulations conceivable 
for him but not imaginable— that does not mean that a four- 
dimensional space can or does exist that is capable, in its turn, 
of bringing our own mathematical conceptions into being in 
concrete form when they transcend our three-dimensional 
world. This would be unduly favoring those who immediately 
interpret the theory of relativity metaphysically. The only aim 
of the artifice we are about to employ is to supply the theory 
with an imaginative prop, so to render it clearer and thus 
make it easier to perceive the errors into which hasty inferences 
would lead us. 

We are therefore simply going to return to the hypothesis 
from which we had set out when we drew two axes at right 
angles and examined a line A'B' on the same plane as they. We 
gave ourselves only the surface of a sheet of paper. This two-di- 
mensional world is endowed by the theory of relativity with an 
additional dimension, which is time: the constant is no longer 
dx 2 + dy 2 but dx 2 + dy 2 - cMP. To be sure, this additional 
dimension is of an altogether special nature, since the constant 
would be dx 2 + dy 2 + dt 2 , without needing an artifice to lead 
it around to this form, if time were a dimension like the others. 
We shall have to keep in mind this characteristc difference, 
with which we have already been occupied and upon which 
we shall soon focus our attention. But we are bypassing it for 
the moment, since the theory of relativity itself invites us to 
do so: if it has had recourse here to an artifice, and posited an 
imaginary time, it was precisely in order that its constant 
might retain the form of a sum of four squares, each with 
unity as coefficient, and in order that the new dimension might 
be provisionally assimilable to the others. Let us therefore 
ask, m a general way, what we bring to, and, what, perhaps, 
we also take away from, a two-dimensional universe when we 
turn us time into an extra dimension. We shall then take ac- 
count of the special role which this new dimension plays in the 
theory of relativity. 

We cannot repeat often enough: the mathematician's time 



is necessarily a time that is measured, and therefore, a spatial- 
ized time. We need not take the position of relativity: from 
any standpoint, mathematical time can be treated as an addi- 
tional dimension of space (we pointed this out more than 
thirty years ago). Let us imagine a surface universe reduced 
to plane P and, on this plane, let us consider a mobile M that 
desaibes any line whatever, for example, a circumference, 
starting at a certain point of origin. We who live in a three- 
dimensional world, will be able to picture this mobile M lead- 
ing a line MN perpendicular to the plane, a line whose chang- 
In g length would at each instant be recording the time elapsed 
from the point of origin. The extremity N of this line will 
describe in the three-dimensional space a curve which, in the 
ase at hand, will be spiral in form. It is easy to see that this 
curve Iaid out in the three-dimensional space yields all the 
temporal details of change in the two-dimensional space P. 
e Stance from any point on the spiral to plane P indicates, 

the^"' moment of time with wnich we are dealing, and 
e tangent to the curve at this point gives us, by its inclina- 
m to plane P, the speed of the moving point at this 
m oment. 3 Thus, it will be thought, the "two-dimensional 

Ciirvp" 4 A T 

b ae uneates only a part of the reality found on plane P 
OndT U " ° nly s ^ ace ' in the P inhabitants' sense of the word, 
real e .°* er nand > the "three-dimensional curve" contains this 
it Mt^u " S entirety: it; has three dimensions of space for us; 
sio i bC tnree " dimen sional space-and-time for a two-dimen- 
visur mathematician "ving on plane P who, incapable of 
lzln S the third dimension, would be led to conceive it 
alyti" n ascertainment of motion, and to express it an- 

sional ^ C ° Uld then learn from us that a three " aimen " 
^ Q CUrve act ually exists as an image. 

feover, once the three-dimensional curve, at once both 

3 A 

-W e 2 e S1D k Ple calcuIation w °uld demonstrate this. 
sioi >al curv " t0 USC these hardl y correct expressions, "two-dimen- 
spiraigj 6 3nd " three -dimensional curve," to refer to the plane and 
""OWicati-, W " There no other way to indicate the spatial and temporal 
^ons of one and the other. 



space and time, has been posited, the two-dimensional curve 
would appear to the mathematician on the flat universe like 
a mere projection onto the plane he inhabits. It would be only 
the surface and spatial aspect of a solid reality which would 
have to be called both time and space. 

In brief, the form of a three-dimensional curve here gives us 
information about both the plane trajectory and the temporal 
details of a motion in two-dimensional space. More generally, 
what is given as motion in a space of any number of dimen- 
sions can be represented as form in a space of one more 

But is this representation really adequate to what is repre- 
sented? Does it contain quite what the latter contains? At first 
glance we might think so, from what we have just said. But 
the truth is that it includes more in one respect, less in an- 
other, and that if the two things appear interchangeable, it is 
because our mind surreptitiously subtracts what is superfluous 
in the representation, and no less surreptitiously inserts what 
is lacking. 

To begin with the second point, it is obvious that becoming, 
properly so called, has been eliminated. This is because science 
has to do with it only in the case at hand. What is its aim? 
Simply to know where the mobile will be at any moment in 
its course. It therefore always betakes itself to the extremity 
of an interval already traversed; it is interested only in the 
result, once that is obtained; if it can portray at one stroke 
every result at every moment, and in such a way as to know 
what result corresponds to what moment, it has achieved the 
same success as the child who has become able to read an 
entire word all at once instead of spelling it letter by letter. 
This is what happens in the case of the point-to-point cor- 
respondence between our circle and spiral. But this corre- 
spondence has meaning only because we mentally traverse the 
curve and occupy points on it successively. If we have been 
able to replace this succession by a juxtaposition, real time by 
a spatialized time, becoming by the become, it is because we 
retain becoming, real duration, within us; when the child 



actually reads a word all at once, he is spelling it virtually 
letter by letter. Let us not therefore imagine that our three- 
dimensional curve gives us, as if crystallized together, the mo- 
tion by which the curve is outlined on the plane and this 
plane curve itself. It has merely extracted from becoming what 
is of interest to science, and science can use this extract only 
because our mind will re-establish the eliminated becoming or 
will feel able to do so. In this sense, the curve of n + 1 dimen- 
sions, already outlined, which would be the equivalent of the 
curve of n dimensions being outlined really represents less 
than it claims to represent. 

But, in another sense, it represents more. Subtracting here, 
adding there, it is doubly inadequate. 

We have obtained it, as a matter of fact, by means of a 
clearly denned operation, through the circular motion, on 
Plane P, 0 f a point M that led the line MN of a length vary- 
ln g with the time elapsed. This plane, circle, line, motion, 
^ese are the completely determinate elements of the operation 
through which the figure was outlined. But the figure all out- 
ined does not necessarily imply this mode of generation. Even 
1 " does imply it, the figure may have been the outcome of the 
m otion of a different line, perpendicular to a different plane, 
w «ose extremity M has described, at quite different speeds, a 
^rve that was not a circumference. Let us, in fact, consider 
an y pIane and project our spiral upon it; the latter will be as 
° earl y representative of the new plane curve, traversed at new 
^ s and amalgamated to new times. If, therefore, in the 
fere 6 described > the s P iral contains less than the circum- 
sens nce and Ae motion we claim to rediscover in it, in another 
e ; 11 c °ntains more; once accepted as the amalgam of a 
am pi ane figure with a certain mode of motion, we can 
tivT^ an infinit y of other P J ane figures in it as well, respec- 
We 6 y COm pleted by an infinity of other motions. In short, as 
hothf n i° UnCed ' this re P resentation is doubly inadequate: it 

f or th' Sh ° rt and gOCS tGO far " And we can thC reason 
pen W By - addin S a dimension to the space in which we hap- 
t0 exist, we can undoubtedly picture a process or a 


becoming, noted in the old space, as a thing in this new space. 
But as we have substituted the completely made for what we 
perceive being made, we have, on the one hand, eliminated 
the becoming inherent in time and, on the other hand, intro- 
duced the possibility of an infinity of other processes through 
which the thing could just as well have been constructed. 
Along the time in which we found the progressive genesis of 
this thing, there was a clearly defined mode of generation; but, 
in the new space, increased by one dimension, in which the 
thing is spread out at one stroke by the joining of time to the 
original space, we are free to imagine an infinity of equally 
pos S1 ble modes of generation; and the one that we have ac- 
tually found, though it alone is real, no longer appears as 
privileged: we shall line it up-wrongly-alongside the others. 

Already we catch a glimpse of the twofold danger to which 
we expose ourselves when we symbolize time by a fourth di- 
mension of space. On the one hand, we risk taking the unfold- 
ing of the whole past, present, and future history of the uni- 
verse for a mere running of our consciousness along this 
history given all at one stroke in eternity; events would no 
longer file before us, it is we who would pass before their 
alignment. And, on the other hand, in the space-and-time or 
space-nme that we shall have thus constituted, we shall believe 
that we are free to choose among an infinity of possible repar- 
titions of space and time. Yet it was out of a well-determined 
space and time that this space-time had been built: only a cer- 
tain special distribution in space and time was real. But we 
make no distmction between it and all other possible distribu- 
tions; or rather, we see no more than an infinity of possible 
distributions, the real distribution being no more than one of 
them. In short, we forget that, measurable time being of neces- 
sity symbolized by space, there is both more and less in this 
space dimension considered as symbol than in time itself. 

But we shall perceive these two points more clearly in the 
u* IT Y C W bCen ima S ini "S a two-dimensional 
7 WH bC Ae indefi *»ely extended plane P. Each 

of the successive states of this universe will be an instantaneous 


image, taking up the whole plane and comprising the totality 
of objects, all flat, of which this universe is made. The plane 
will therefore be like a screen upon which the cinematography 
of the universe would be run off, with the difference however 
that here there is no cinematography external to the screen, 
no photography projected from without; the image takes form 
on the screen spontaneously. Now, the inhabitants of plane P 
will be able to imagine the succession of cinematographic 
Mages in their space in two different ways. They will split 
into two camps, depending upon whether they adhere more 
to the data of experience or to the symbolism of science. 

The first will be of the opinion that there really are suc- 
cessive images, but not all lined up on a roll of film; and this, 
j* two reasons: (1) Where would the film be housed? By 
^thesis, each of the images, covering the screen by itself, 
Th^ ° f a perha P s infinit e space, that of the universe. 

es e images therefore really have no alternative but to exist 
^successively; they cannot be given globally. Besides, time 
cessio PreSentS ltseIf t0 our consciousness as duration and suc- 
juxtapo ^. tributes irre <iucible to any other and distinct from 
nuned T' ^ ° n a fiIm ' evei 7 th ing would be predeter- 
be our ^ Prefer ' determined - Illusory, therefore, would 
m ccessioT SC H OUSneS - ° f choosin S' actin S' creating. If there is 
its way n h duration ' il is on ly because reality hesitates, feels 
sur e, tne h UaUy W ° rkS ° Ut the unfor eseeable novelty. To be 
Peat; this C ° f absolute determination in the universe is 
Bu t what*- 18 CXactly why a mathematical physics is possible. 
d «res onlY 1 th Predete - rmined is virtuall y alread y mad « and en- 
^ what " r ° Ugh US connec ti° n with what is in the making, 
""ttweavin 8 ^ duration and succession; we must take this 
future h- mt ° aCC0Unt and then see that the past, present, 
a ro11 of film 1 a 017 ° £ the universe cannot be given globally on 


pothers would reply: "In the first place, we have nothing 

'"''^^oTft^.h P 0 ' 111, t0 what we o^ed "the cinematographic 
^ m ^m^'"f d ^ reference to our cinematographic repre- 
ss, see L Evolution criatrice (Creative Evolution), Chap. IV. 



to do with your so-called unforeseeableness. The aim of sci- 
ence is to calculate and therefore to foresee; we shall therefore 
disregard your feeling of indeterminacy, which is perhaps only 
an illusion. Now, you say that there is no room in the universe 
to house images other than the image designated as present. 
This would be true if the universe were doomed to having 
only two dimensions. But we can imagine a third to which our 
senses cannot attain and across which our consciousness would 
travel when unfolding in "time." Thanks to this third dimen- 
sion of space, all the images making up all the past and future 
moments of the universe are given at one stroke along with 
the present image, not laid out with respect to one another 
like frames on a roll of film (for that, indeed, there would be 
no room), but arranged in a different order, which we do not 
succeed in imagining, but which we can nevertheless conceive. 
To live in time consists in traversing this third dimension, 
that is, in itemizing it, in perceiving one by one the images 
that it enables to be juxtaposed. The apparent indeterminate- 
ness of what we are about to perceive lies merely in the fact 
that it has not yet been perceived; it is an objectivizing of our 
ignorance. 8 We believe that images are created in so far as 
they appear, precisely because they seem to appear to us, that 
is, to arise before us and for us, to come toward us. But let us 
not forget that all motion is reciprocal or relative: if we per- 
ceive them coming toward us, it is also true to say that we are 
going toward them. They are there in reality; lined up, they 
await us; we march past them. Let us not say, therefore, that 
events or accidents befall us; it is we who befall them. And we 
would immediately ascertain this if we were as acquainted 
with the third dimension as with the others." 

I shall now imagine that I have been appointed arbitrator 
between the two camps. Turning to those who have just 
spoken, I would say to them: "Let me first congratulate you 
upon having only two dimensions, for you are thus going to 

• In the pages devoted to the "cinematographic mechanism of thought," 
we once showed that this way of reasoning is natural to the human mind 



obtain for your thesis a proof for which I would vainly seek, 
were I to pursue an argument analogous to yours in the space 
into which fate has thrust me. I happen, as a matter of fact, 
to live in a three-dimensional space; and when I agree with 
some philosophers that it can really have a fourth, I am saying 
something that is perhaps absurd in itself, although mathe- 
matically conceivable. A superman, whom I would appoint, 
in my turn, as arbitrator between them and me would perhaps 
explain that the idea of a fourth dimension is obtained 
fcough the extension of certain mathematical habits con- 
tracted in our space (entirely as you obtained the idea of 
a third dimension), but that this time the idea does not and 
cannot correspond to any reality. There is, nevertheless, a 
three-dimensional space, where I happen to be: this is a good 
thing for you, and I shall be able to give you information. 
Yes, you have guessed right in believing that the coexistence of 
images like yours, each extending over an infinite 'surface,' 
» possible when it is impossible in the truncated space where 
your whole universe appears to you to abide at each instant. 
It is enough that these images-which we call 'flat'-pile up, 
w e say, one on top of the other. There they are, all piled 
U P- 1 see your 'solid,' as we call it, universe; it is made of the 
P'hng up of all your flat images, past, present, and future. 

also see your consciousness traveling perpendicularly to these 
superimposed 'planes,' never taking cognizance of any but the 
°ne it crosses, perceiving it as the present, then remembering 
jhe one it leaves behind, but ignorant of those which are in 
fr °nt and which enter its present, one at a time, forthwith 
enriching i ts past . 

j* 1 ". this is what strikes me further. 
. 1 have taken random images, or rather pellicles without 
! mages on them, to represent your future, which I do not 
u now " 1 have thus piled up on top of the present state of your 

" lv erse future states that remain blank for me; they form a 
Pedant to the past states on the other side of the present 
b e ' whic h past states I perceive as definite images. But I am 

y no mea ns sure that your future coexists in this way with 



your present. It is you who are telling me it does. I have drawn 
my figure to your specifications, but your hypothesis remains 
an hypothesis. Do not forget that it is an hypothesis and that 
it merely expresses certain properties of a very special class of 
events, carved out of the immensity of the real, with which 
physical science is occupied. Now, I can tell you, letting you 
benefit from my experience of the third dimension, that your 
representation of time by space is going to give you both more 
and less than you wish to represent. 

"It will give you less, because the heap of piled-up images 
comprising every state of the universe contains nothing that 
either implies or explains the motion by which your space P 
invests them one at a time, or by which (it amounts to the 
same thing, according to you), one at a time, they come to 
fill the space P where you are. I am well aware that, in your 
eyes, this motion is of no consequence. Since all the images 
are given virtually— and this is your conviction— since we are 
theoretically in a position to take the one we want out of the 
front part of the pile (in this lies the calculation or prevision 
of an event), the motion that would oblige you first to pass 
along images lying between that one and the present image- 
the motion that would actually be time-seems to you a mere 
'delay' or hindrance brought to bear, in actuality, upon a per- 
ception that, by right, is immediate; there would be here only 
a deficiency in your empirical knowledge, exactly made up for 
by your mathematical science. In a word, it would be some- 
thing negative; and we would not be claiming more, but less 
than we had, when we posit a succession, that is, a necessity 
for leafing through the album, when all the leaves are there. 
But I, who experience this three-dimensional universe and 
can there actually perceive the motion imagined by you, I 
must mform you that you are looking at only one aspect of 
mobility and, consequently, of duration; the other, essential, 
one escapes you. We can, no doubt, consider every part of 
every future, predetermined state of the universe as theoret- 
ically piled up one on top of the other, and logically given in 
advance; we only express their predetermination in this way. 


But these parts, constitutive of what we call the physical world, 
are framed in others upon which your calculation has until 
now had no hold and which you declare calculable as the 
result of an entirely hypothetical assimilation; these are the 
organic, the conscious. I, who have been inserted into the 
organic world through my body, and into the world of con- 
sciousness through my mind, I perceive its forward progress as 
a gradual enrichment, a continuity of invention and creation. 
For me, time is what is most real and necessary; it is the 
fundamental condition of action— what am I saying?-it is 
action itself; and my obligation to live it, the impossibility of 
ever encroaching upon the coming interval of time, would be 
enough to show me— if I did not have it as an immediate ex- 
perience-that the future is really open, unforeseen, indetermi- 
nate. Do not consider me a metaphysician, if you thus refer to 
4e man of dialectical constructions. I have constructed noth- 
in g- I have merely noted. I am confiding to you what greets 
senses and consciousness: what is immediately given must 
considered real as long as we have not convicted it of being 
a mere a Ppearance; if you see it as illusory, it is up to you to 
P r °ve this. But you suspect it as illusory only because you 
yourself are creating a metaphysical construction. Or, rather, 
e instruction has already been created; it dates from Plato, 
w o held time to be a mere deprivation of eternity; and most 
ancient and modern metaphysicians have adopted it just as 
•stands, because it does, in fact, answer a fundamental need 
° human understanding. Made to establish laws, that is, to 
Q{ ract cer tain unchanging relations from the changing flux 
things, our understanding is naturally inclined to see only 
a em; the 7 alone exist for it; it therefore fulfills its function, 
tim* 618 US P ur P ose > in taking up a position outside of the 
be 6 that fl °ws and endures. But the mind, which extends 
tial ShCer understandin g. is well aware that, if the essen- 
that Work of intelligence is the extraction of laws, it is in order 

our ^ aCti ° n may know what to take into account ' so that 

treatd haVC a better grip on thin S s: the " nderstandin & 
s duration as a deficiency, a pure negation, in order that 



we may be able to work with the greatest possible efficiency 
within this duration, which is, however, what is most positive 
in the world. The metaphysics of most metaphysicians is there- 
fore only the very law of the functioning of the understanding, 
which is one of the faculties of mind, but not mind itself. The 
latter, in its integrality, takes account of integral experience; 
and the integrality of our experience is duration. Hence, no 
matter what you do, you eliminate something, even what is 
essential, in replacing the singly passing states of the universe 
by a block universe posited once and for all. 7 

"You are thereby claiming less than you should. But, in 
another sense, you are claiming more. 

"You are, in fact, convinced that your plane P passes through 
every image, ready and waiting for you, of all the successive 
moments of the universe. Or— what amounts to the same 
thing— you are convinced that each of these images given in 
the instantaneous or in eternity has been doomed, by reason 
of a weakness in your perception, to seem to you to be passing 
onto your plane P one at a time. It makes little difference, 
moreover, whether you express yourself in one way or the 
other; in both cases there is a plane P— this is space— and a 
shift of this plane in a direction parallel to itself— this is time— 
which makes the plane traverse the totality of the once-and-for- 
all posited block. But if the block is really given, you can just 
as easily intersect it by any other plane P' again moving paral- 
lel to itself and thus traversing the totality of the real in a 
different direction. 8 You will have effected a new distribution 

Tin L'Evolution criatrice (Creative Evolution), Chap. IV, we dwelled 
at length upon the connection established by metaphysicians between the 
block and the images given one at a time. 

8 It is true that, in our usual conception of spatialized time, we have 
never tried to shift the direction of time in actual fact, and to imagine a 
new distribution of the four-dimensional space-time continuum: it would 
offer no advantage and give incoherent results, whereas this operation 
seems to force itself upon us in the theory of relativity. Still, as we see it, 
the amalgam of time with space, which we claim to be characteristic of 
this theory, is, strictly speaking, conceivable in the everyday theory, even 
though it may look different there. 


of space and time just as legitimate as the first, since only the 
solid block has absolute reality. In fact, such is actually your 
hypothesis. You imagine that, by adding an extra dimension, 
you have obtained a three-dimensional space-time that can be 
divided into space and time in an infinite number of ways; 
yours, the one you experience, would be only one of them; it 
would rank with the others. But I, who see what all these ex- 
periences of observers attached to and moving with your P' 
planes would be, experiences which you merely imagine, I can 
inform you that, having the vision of an image composed of 
points borrowed from all the real moments in the universe, 
they would live in incoherence and absurdity. The aggregate 
of these incoherent and absurd images does, indeed, reproduce 
the block, but it is only because the block has been constituted 
in quite another manner— by a particular plane moving in a 
particular direction-that a block exists at all, and that we can 
play about with the fantasy of mentally reconstituting it by 
means of any plane at all moving in some other direction. To 
rank these fantasies with reality, to say that the motion which 
is actually productive of the block is only one of a number of 
Possible motions, is to disregard the second point to which I 
just drew your attention: in the block which is ready-made and 
set h" ee of the duration where it was being made, the result, 
once obtained and cut off, no longer bears the clear stamp of 
the work by which we obtained it. A thousand different men- 
ki operations, would just as easily recompose it in idea, even 
"tough it has really been composed in a certain unique way. 
After the house has been built, our imagination can roam all 
0ver 11 an <i rebuild it just as easily by first setting the roof, 
a " d ^ hitching the stories to it, one at a time. Who would 
P ac e this method on the same footing with that of the archi- 
tect and consider both equivalent? Looking closely, we see that 
* architect's method is the only effective way to compose the 
°le, that is, to make it; the others, despite appearances, are 
J y ways to decompose it, that is, in short, to unmake it; there 
st > then, as many of these ways as we like. What could be 



built only in a certain order can be demolished any which 

Such are the two points we must never lose sight of when 
we join time to space by endowing the latter with an extra 
dimension. We have taken the most general case; we have not 
yet considered the very special look of this new dimension in 
the theory of relativity. This is because every time the theo- 
reticians of relativity leave pure science to give us an idea of 
the metaphysical reality which that mathematics expresses, 
they begin by implicitly allowing the fourth dimension at least 
the attributes of the other three, even bringing in some- 
thing more. In talking about their space-time, they take the 
following two points for granted: (1) Every partitioning of it 
in space and time must be accorded equal rank (it is true that 
in the hypothesis of relativity, these partitionings can only be 
made according to a special law, to which we shall soon recur); 
(2) our experience of successive events only illumines, one by 
one, the points of a line given all at once. They seem not to 
have realized that the mathematical expression of time, neces- 
sarily imparting to it, in effect, the characteristics of space and 
requiring that the fourth dimension, whatever its own quali- 
ties, first have those of the other three, will sin both by excess 
and deficiency, as we have just shown. Whoever does not pro- 
vide a corrective here runs the risk of mistaking the philo- 
sophical meaning of the theory of relativity and of giving a 
mathematical representation the status of a transcendent real- 
ity. We shall be persuaded of this by repairing to certain 
passages in Eddington's already classic volume: "Events do not 
happen; they are there and we meet them on our way. The 
'formality of taking place' is merely an indication that the 
observer, in his voyage of exploration, has passed into the 
absolute future of the event in question, and is of no great 
significance." » Before that, we read in one of the first works 
on the theory of relativity, by Silberstein, 10 that Wells had 

9 Arthur S. Eddington, Space, Time and Gravitation (Cambridge: Cam- 
bridge University Press, 1920), p. 151. 

lOLudwik Silberstein, The Theory of Relativity (London: MacmiUan 
and Co., Ltd., 1914), p. 134. 



wondrously anticipated this theory when he had his "time- 
traveler" say that "there is no difference between time and 
space except that our consciousness moves along time." 

But we must now turn our attention to the special look 
which the fourth dimension takes on in the space-time of Min- 
kowski and Einstein. Here, the constant ds 2 is no longer a sum 
of four squares, each having the coefficient of unity, as it would 
be if time were a dimension like the others: the fourth square, 
assigned the coefficient c 2 , must be subtracted from the sum of 
the preceding three, and thus proves a case apart. We can 
smooth out this singularity of mathematical expression by a 
suitable artifice; it nonetheless remains in the thing expressed 
and the mathematician advises us of this by saying that the 
first three dimensions are "real" and the fourth, "imaginary." 
Let us examine this special form of space-time as closely as 

But let us at once announce the result toward which we are 
heading. It will necessarily resemble greatly the one that our 
inquiry into multiple times gave us; it can, indeed, be only a 
new expression of it. Against common sense and the philo- 
sophic tradition, which declare for a single time, the theory of 
relativity had first appeared to assert the plurality of times. On 
doser inspection, we had never found more than a single real 
tone, that of the physicist engaged in building up his science; 
*e others are virtual, that is, imaginary times, attributed by 
h "n to virtual, that is, phantasmal observers. Each of these 
Phantasmal observers, suddenly coming to life, would install 
himself in the real duration of the former real observer, who 
w °uld become phantasmal in his turn. Thus, the usual idea 
° real time quite naturally continues to hold good with, in 
Edition, a mental construction intended to represent how, if 
one applies the Lorentz equations, the mathematical expres- 
Sl °n of electromagnetic facts remains the same for the observer 
considered motionless and for the observer to whom any uni- 
°"n motion at all is attributed. Now, Minkowski's and Ein- 
J s P a ce-time represents nothing else. If by four-dimensional 
P a ce-time we understand a real environment in which real 



beings and objects evolve, the space-time of the theory of rela- 
tivity is everyone's, for we all make the vague gesture of posit- 
ing a four-dimensional space-time as soon as we spatialize time; 
and we cannot measure time, we cannot even talk about it, 
without spatializing it. 11 But, in this space-time, time and 
space remain separate; space can neither disgorge time nor 
time recede into space. If they bite into one another, in pro- 
portions varying with the speed of the system (this is what they 
do in Einstein's space-time), then we are no longer dealing 
with anything more than a virtual space-time, that of a physi- 
cist imagined as experimenting and no longer that of the 
physicist who does experiment. For this latter, space-time is at 
rest, and, in a space-time at rest, time and space remain sepa- 
rate; they intermingle, as we shall see, only in the mixing 
produced by the system's motion; but the system is in motion 
only if the physicist who happened to be there abandons it. 
Now, he cannot abandon it without installing himself in an- 
other system; the latter, which is then at rest, will have a space 
and a time as clearly separated as ours. So that a space that 
swallows time, and a time that, in turn, absorbs space, are a 
time or a space always virtual and merely imagined, never real 
and experienced. It is true that the conception of this space- 
time will then influence the perception of actual space and 
time. Across the time and space we had always known to be 
separate and, for that very reason, structureless, we shall per- 
ceive, as through a transparency, an articulated space-time 
structure. The mathematical notation of these articulations, 
carried out upon the virtual and brought to its highest level 
of generality, will give us an unexpected grip on the real. We 
shall have a powerful means of investigation at hand, a prin- 
ciple of research, which, we can predict, will not henceforth 
be renounced by the mind of man, even if experiment should 
impose a new form upon the theory of relativity. 

"This is what we expressed in another form (pp. 57ff.) when we said 
that saence has no way of distinguishing between time unfolding and time 
unfolded. It spatializes it by the very fact that it measures it. 


To show how time and space begin to interweave only when 
both become fictional, let us return to our system S' and to 
our observer who, actually located in S', mentally transfers to 
a different system S, immobilizes it, and then imagines S' en- 
dowed with every possible speed. We wish to find out the more 
special meaning, in the theory of relativity, of the interweav- 
ing of space with time considered as an additional dimension. 
We shall not be changing anything in the outcome and shall 
be simplifying our exposition, by imagining that the space of 
systems S and S' has been reduced to a single dimension, a 
straight line, and that a worm-shaped observer in S' inhabits 
part of this line. Basically, we are only getting back to the 
situation prevailing a while back (p. 128). We said that as long 
as our observer keeps thinking in S' where he is, he purely and 
simply notes the persistence of length A'B' designated by I. 
But, as soon as he mentally transfers to S, he forgets the estab- 
lished, concrete invariability of length A'B' or of its square P; 
he conceives it only in abstract form as the invariance of a dif- 
ference between two squares L 2 and c 2 T 2 , which would alone 

be given (calling L the lengthened space J—j > and T the 
interval of time 1 . ~ which has come to be intercalated 

betwen the two events A' and B', perceived inside system S' as 
simultaneous). We who know spaces of more than one dimen- 
sion, have no trouble in geometrically conveying the difference 
between these two conceptions; for, in the two-dimensional 
s Pace that for us surrounds line A'B' we have but to erect on 
*e latter a perpendicular B'C equal to cT, to discern at once 
Aat the real observer in S' really perceives side A'B' of the right 
Wangle as invariable, while the fictional observer in S directly 
Perceives (or, rather, conceives) only the other side B'C and 
tte hypotenuse A'C of this triangle: line A'B' would then be 
n ° more for him than a mental outline by which he completes 



the triangle, an expression represented by \] A'C' 2 - B'C' 2 . Now, 
suppose that the wave of a magic wand places our observer, 
real in S' and fictional in S, in circumstances like ours and 
allows him to perceive or conceive a space of one more dimen- 
sion. As a real observer in S', he will perceive the straight line 
A'B'; this is the real. As an imaginary physicist in S, he will 
perceive or conceive the broken line A'C'B'; this is only the 
virtual; it is the straight line A'B' appearing lengthened and 
undoubled in the mirror of motion. Now, the straight line A'B' 
is space. But the broken line A'C'B' is space and time; and so 
would be an infinity of other broken lines A'D'B', A'E'B', etc., 
corresponding to different speeds of system S', while line A'B' 
remains space. These broken, merely virtual, lines of space- 
time come out of the straight line of space only because of the 
motion that the mind imparts to the system. They are all sub- 
ject to the law that the square of their space part, diminished 
by the square of their time part (we have agreed to make the 
speed of light our unit of time) leaves a remainder equal to 
the invariable square of the straight line A'B', the latter a line 
of pure space, but real. Thus, we see exactly the relation of 
the space-time amalgam to the separate space and time, which 
we had always left side by side even though we had made an 
additional dimension of space out of time by spatializing it. 
This relation becomes quite striking in the particular case we 
have chosen by design, the one in which line A'B', perceived 
by an observer situated in S', joins two events A' and B' given 
m this system as simultaneous. Here, time and space are so 
clearly separate that time is eclipsed, leaving only space; a 
space A'B', this is all that is clearly noted, this is the real. But 
this reality can be reconstituted virtually by an amalgam of 
virtual space and virtual time, this space and time lengthening 
with every increase in the virtual speed imparted to the system 
by the observer who ideally detaches himself from it. We thus 
obtain an infinity of merely mental space and time amalgams, 
all equivalent to space pure and simple, perceived and real. 

But, the essence of the theory of relativity is to rank the real 
vision with the virtual visions. The real would be only a spe- 



cial case of the virtual. There would be no difference in kind 
between the perception of the straight line A'B' in system S', 
and the conception of the broken line A'C'B', when we im- 
agine ourselves in system S. The straight line A'B' would be a 
broken line like A'C'B' with a null segment C'B', the value 
zero assumed here by c 2 T 2 being a value like the others. Mathe- 
matician and physicist certainly have the right to express them- 
selves in this way. But the philosopher, who must distinguish 
between the real and the symbolic, will speak differently. He 
will merely describe what has just happened. There is a real, 
perceived length A'B'. And if we agree to claim only that, con- 
sidering A' and B' instantaneous and simultaneous, we simply 
have, by hypothesis, that length of space plus a nothing of 
time. But a motion mentally imparted to the system makes the 
originally considered space appear time-inflated: I 2 becomes L 2 , 
that is, l 2 + c 2 T 2 . The new space will then have to disgorge 
time, and L 2 will have to be reduced by c 2 T 2 before we can 
find again. 

We are thus brought back again to our previous conclusions. 
We were shown that two events, simultaneous for an individ- 
ual observing them inside his system, are successive for an out- 
sider imagining it in motion. We granted this, but pointed out 
that despite our giving the name of time to the interval be- 
tween the two events become successive, it cannot harbor any 
event. It i S) we said, "expanded out of nothing." 12 Here we are 
witnessing this expansion. For the observer in S', the distance 
between A' and B' was a length of space I augmented by a zero 
°f time. When the reality Z 2 becomes the virtuality U, the zero 
°t real time blossoms into a virtual time c^T 2 . But this inter- 
val of V i r t ua i time is only the nothing of the original time, 
Producing some kind of optical effect in the mirror of motion.. 
Thought can no more lodge even the most fleeting event in it, 
*an we can move a piece of furniture into a room perceived 
ln the depths of a mirror. . 

fi ut we have been looking at a special case, the one m whicn 
th « events A' and B' are, from within system S', perceived as 

"See above, p. 106. 



simultaneous. This seemed the best way to analyze the opera- 
tion by which space is added to time, and time to space, in the 
theory of relativity. Let us now take the more general case in 
which events A' and B' occur at different moments for the ob- 
server in S'. We return to our original notation: we shall call 
t\ the time of event A', and f' 2 that of event B'; we shall desig- 
nate by x' 2 - x\ the distance in space from A' to B', x' 2 and x\ 
being the respective distances from A' and from B' to a point 
of origin O'. To simplify things, we shall again imagine space 
reduced to a single dimension. But this time we shall ask our- 
selves how the observer inside S', finding in this system both 
the constancy of the x' 2 - x\ space length and that of the 
t' 2 -t\ time length for any imaginable speed of this system, 
would picture this constancy when mentally entering a motion- 
less system S. We know 13 that (x' 2 - x\) 2 would thereupon 
have to be expanded into 

-L-[(x' 2 _*'i)+"('' 2 -<'i)] 2 

c 2 

a quantity that exceeds (x' 2 - x^) 2 by 

-^5 ( x '2 ~ *'i) 2 + ^Ca - *'i) 2 + 2*(x' 2 - x',) (r 2 - t' x ) j . 
c 2 

Here again, as we see, a time would have come to inflate a 

But, in its turn, a space has been added onto a time, because 
what was originally (t' 2 - t\) 2 has become. 14 


a quantity that exceeds (f 2 - t\y by 

- +5('' 2 - W ^ (x\ - *' x ) (P, - 1\) ] . 

c 2 

The result is that the square of time has been increased by a 
is See p. ISO. 
"See p. 181. 


quantity which, multiplied by c 2 , would give the increase in 
the square of space. Thus, with space gathering up time and 
time gathering up space, we see the invariance of the differ- 
ence (x 2 - Xj)2 - c 2 (« 2 - tj) 2 forming before our very eyes for 
any assigned speed of the system. 

But this amalgam of space and time comes into being for 
the observer in S' only at the exact instant that he mentally 
sets the system in motion. And the amalgam exists only in his 
mind. What is real, that is, observed or observable, is the 
separate space and time with which he deals in his system. 
He can associate them in a four-dimensional continuum; this 
we all do, more or less confusedly, when we spatialize time, 
and we spatialize it as soon as we measure it. But space and 
time then remain separately invariant. They amalgamate or, 
more precisely, their invariance is transferred to the difference 
(*2-*i) a - 02(^-^)2 only f or our phantasmal observers. The 
real observer will offer no objection, for he remains wholly 
unaffected: as each of his terms x 2 - x x and t 2 - h, space inter- 
val and time interval, is invariable, from whatever point he 
considers them inside his system, he abandons them to the 
Phantasmal observer so that the latter may have them enter 
as he pleases into the expression of his invariant; he adopts 
expression beforehand, he knows in advance that it will 
fit his system as he himself envisages it, for a relation between 
instant terms is necessarily constant. And much is gained, for 
Ae expression with which we provide him is that of a new 
Physical truth: it points out how the "transmission" of light 
behaves with regard to the "translation" of bodies. 

But w hile it informs him of the relation of the transmission 
J° Ae translation, it tells him nothing new about space and 

lme; the latter remain what they were, separate from one an- 
er - ^capable of mingling except as the result of a mathe- 
matical fiction intended to symbolize a truth in physics. For 
™* space and time which interpenetrate are not the space 

? taie of any physicist, real or conceived as such. The real 
Ptysicist makes his measurements in the system in which he 
^ ^elf. and which he immobilizes by adopting it as his 



system of reference; time and space there remain separate and 
mutually inpenetrable. Space and time interpenetrate in mov- 
ing systems in which the real physicist does not exist, in which 
there live only physicists imagined by him— imagined for the 
greater good of science. But these physicists are not imagined 
as real or able to be so; to suppose them real, to attribute a 
consciousness to them, would be to give their system the status 
of a system of reference, to transport oneself there and become 
identical with them, to declare that their time and space have 
ceased to interpenetrate. 

We thus return by a long detour to our starting point. We 
are merely repeating, for space convertible into time and for 
time reconvertible into space, what we had said about the 
plurality of times, and about succession and simultaneity con- 
sidered as interchangeable. And this is quite natural, since we 
are dealing with the same thing in both cases. The invariance of 
the expression dx 2 + dy 2 + dz 2 - c 2 df 2 follows immediately from 
the Lorentz equations. And the space-time of Minkowski and 
Einstein only symbolizes this invariance, as the hypothesis of 
multiple times and simultaneities convertible into successions 
only interprets these equations. 


Time in Special Relativity and Space in 
General Relativity 

We are now at the end of our study. It had to bear upon time 
and the paradoxes of time, which we usually associate with the 
weory of relativity. Hence it is confined to special relativity. 
Are we therefore left in the abstract? Not at all, nor would we 
J™* "aytWng essential to add regarding time, if we intro- 

whih 3 gravitational field into the simplified reality with 
toch we have been occupied until now. Indeed, according to 
theory of general relativity, we can no longer either define 
^synchronization of clocks or declare the speed of light con- 
ti * ! n a gravitational field. In all strictness, therefore, the 
P ca definition of time would vanish. As soon as we wish to 
to th meaning t0 the " time " co-ordinate, we necessarily submit 
in *h ?>* iltio ™ of special relativity, going to look for them 
w 4emfinite, if necessary. 

4e mstant ' a univer se of special relativity is tangent to 
consider 61 * 6 ° f general relativit Y- Moreover, we never have to 
fields of SPCedS Com P arable t0 that of ^ght. or gravitational 
* a suffi P ^° POrti0nal intensit y- Therefore we can in general, 
special rel* 6111 - approximation > borrow the notion of time in 
^e is r f tlVUy 3nd retain il J ust as il stands - In this sense » 
relativity 6 ^ l ° special rel ativity, as space is to general 

^ relativit 1116 ° f special relativit Y and the space of gen- 
A careful ^ * r ° m bav ' n g tne same degree of reality, 

^thenhl ° f th " P ° int would be singularly instructive 

*• *e oiII^ Pher " 11 WOuld bear out ±e radical ^s^ 11 ^ 011 
*« drew between the nature of real time and pure 




space, improperly considered analogous by traditional philoso- 
phy. And it would perhaps not be without interest for the 
physicist. It would reveal that the theory of special relativity 
and that of general relativity are not animated by exactly the 
same spirit and do not have quite the same meaning. The first, 
it must be added, has sprung from a collective effort, while the 
second reflects Einstein's own genius. The former provides us, 
above all, with a new formula for results already obtained; 
it is truly a theory, in the literal sense of the word, a way of 
viewing. The latter is essentially a method of investigation, an 
instrument of discovery. But we need not enter into their com- 
parison. Let us merely touch upon the difference between time 
in one and space in the other. This will be to return to an 
idea often expressed in the course of the present essay. 

When the physicist of general relativity determines the 
structure of space in general relativity, he is referring to a 
space in which he is actually located. He checks every propo- 
sition he puts forward with appropriate measuring devices. 
The portion of space whose curvature he describes may be ever 
so remote: theoretically he would transport himself there, 
would have us witness the verification of his formula. In short, 
the space of general relativity presents details that are not 
merely conceived but could be perceived as well. They relate 
to the system in which the physicist lives. 

But, in the theory of special relativity, the details of time 
and, more particularly, the plurality of times, do not merely 
escape, in actual fact, the observation of the physicist who 
posits them: they are unverifiable in principle. While the 
space of general relativity is a space in which we exist, the 
times of special relativity are so defined as to be, all but one, 
times in which we do not exist. We cannot be in them, because 
we bring with us, wherever we go, a time that chases out the 
others, just as a pedestrian's lamp rolls back the fog at each 
step. We do not even conceive ourselves as being in them, 
because to enter one of these expanded times mentally would 
be to adopt the system to which it belongs, to make it our 
system of reference; at once this time would contract and again 



become the time that we live inside a system, the time that we 
have no reason for not believing to be the same in every 

Expanded and broken-up times are therefore auxiliary times, 
intercalated by the physicist's mind between the start of his 
calculations, which is real time, and its finish, which is still 
this same real time. In the latter we have made the measure- 
ments with which we operate; to the latter do the operation's 
results apply. The others are intermediary between the state- 
ment and solution of the problem. 

The physicist puts them all on the same plane, gives them 
the same name, treats them in the same way. And he is justi- 
fied in this. All are, in fact, measurements of time; and as the 
measurement of a thing is, in the eyes of the physicist, that 
very thing, they must all be times for the physicist. But in only 
one of them-we believe we have demonstrated this-is there 
succession. Consequently, only one of them endures; the others 
do not - While the former is a time unquestionably placed back 
to back with the length that measures it, but is separate from 
the others are only lengths. More precisely, the former is 

°th a time and a "light-line"; the others are only light-lines. 

ut as these last arise from a lengthening of the former, and, 
f * e first w as pasted to time, we think of them as lengthened 
times. Whence comes the infinite number of times in special 
re ^ tmt y- This plurality, far from ruling out the oneness of 
realtime, presupposes it. 

he paradox begins when we assert that all these times are 
0r a " ies " that is, things perceived or able to be perceived, lived 
for i l ° be lived " We had im P licitl y assumed the opposite 
with ° f . them ~ exce Pt one-when we had identified time 
^ the light-line. Such is the contradiction that our mind 

be CVen When il does not P erceive 11 clearlv - Nor ' il must 
or/ • iS il attr ibutable to any physicist as such: it arises 
tiorl m 3 ? hysics P osin g as a metaphysics. To this contradic- 
its r mind cannot adjust. We have been wrong to attribute 
0r a e t Slstan ce to a prejudice of common sense. Prejudices vanish 
least weaken upon reflection. But, in the present case, 



reflection strengthens our conviction and even ends by render- 
ing it unshakable, because it reveals in the times of special 
relativity— one among them excepted— times without duration, 
in which events cannot succeed each other, nor things subsist, 
nor beings age. 

Aging and duration belong to the order of quality. No work 
of analysis can resolve them into pure quantity. Here the thing 
remains separate from its measurement, which besides, bears 
upon a space representative of time rather than upon time 
itself. But it is quite otherwise with space. Its measurement 
exhausts its essence. This time, the details discovered and de- 
scribed by physics belong to the thing and no longer to a 
mental view of it. Let us rather say, they are reality itself; the 
thing is, this time, relation. Descartes reduced matter— consid- 
ered at the instant— to extension; physics, in his eyes, attained 
to the real insofar as it was geometrical. A study of general 
relativity, parallel to the one we have made of special relativity, 
would show that the reduction of gravitation to inertia has 
justly been an elimination of ready-made concepts which, 
coming between the physicist and his object, between the mind 
and the relations constitutive of the thing, was at this point 
preventing physics from being a geometry. In this respect, Ein- 
stein is the continuator of Descartes. 



The Journey in the Projectile 

We have stated but cannot repeat often enough: in the theory 
of relativity, the slowing of clocks is only as real as the shrink- 
ing of objects by distance. The shrinking of receding objects 
« the way the eye takes note of their recession. The slowing 
°f the clock in motion is the way the theory of relativity takes 
note of its motion: this slowing measures the difference, or 

distance," in speed between the speed of the moving system 
to which the clock is attached and the speed, assumed to be 
zer o, of the system of reference, which is motionless by defini- 
ll on; it is a perspective effect. Just as upon reaching a distant 
°°ject we see it in its true size and then see shrink the object 
we have just left, so the physicist, going from system to system, 

1 always find the same real time in the systems in which he 
"Walls himself and which, by that very fact, he immobilizes, 
, Ut Wl11 always, in keeping with the perspective of relativity, 

* ve to attribute more or less slowed times to the systems 
^ ch he va cates, and which, by that very fact, he sets in mo- 
10n at greater or lesser speeds. Now, if I reasoned about some- 
° n |- far away, whom distance has reduced to the size of a 

' get, as about a genuine midget, that is, as about someone 
co ° " and acts Iike a midget, I would end in paradoxes or 

ntradictions; as a midget, he is "phantasmal," the shortening 
N | fi g ure being only an indication of his distance from me. 
ide l paradoxical will be the results if I give to the wholly 
inth Phantasmal clock th at tells time in the moving system 
« perspective of relativity, the status of a real clock telling 
^ s time to a real observer. My distantly-removed individuals 
thatch 6nough and > as reaJ . reta in their size; it is as midgets 
th ey are phantasmal. In the same way, the clocks that 



shift with respect to motionless me are indeed real clocks; but 
insofar as they are real, they run like mine and tell the same 
time as mine; it is insofar as they run more slowly and tell a 
different time that they become phantasmal, like people who 
have degenerated into midgets. 

Let us imagine a normal-sized Peter and Paul conversing. 
Peter stays where he is, next to me; I see him and he sees him- 
self in his true size. But Paul moves off and becomes midget- 
sized in Peter's eyes and mine. If I now go around thinking 
of Peter as normal-sized and of Paul as a midget, picturing 
him that way back with Peter and resuming his conversation, 
I shall necessarily end in absurdities or paradoxes; I have no 
right to bring Peter, who has remained normal, in contact 
with Paul turned midget, to imagine that the latter can speak 
with the former, see him, listen to him, perform any action at 
all, because Paul, as midget, is only a mental view, an image, 
a phantom. Nevertheless, this is exactly what both partisan 
and adversary of the theory of relativity did in the debate, 
begun at the College de France in April 1922, on the implica- 
tions of special relativity. 1 The former merely kept pointing 
to the perfect mathematical coherence of the theory, but then 
retained the paradox of multiple and real times— as if one were 
to say that Paul, having returned to the vicinity of Peter, had 
been changed into a midget. The latter probably wanted 
no paradox, but he could have avoided it only by showing that 
Peter is a real being and that Paul turned midget is a mere 
phantom, that is, by making a distinction that belongs no 
longer to mathematical physics but to philosophy. Remaining, 
on the contrary, on his opponents' ground, he only succeeded 
in furnishing them with an occasion for reinforcing their posi- 
tion and confirming the paradox. The truth is that the para- 
dox vanishes when we make the distinction that is indispensa- 
ble. The theory of relativity remains intact, with its infinite 
multiplicity of imaginary times and a single, real time. 

This is exactly our argument. That there has been some 

i We are alluding to an objection to the theory of relativity voiced by 
M. Painleve. 


difficulty in grasping it, and that it is not always easy, even 
for the relativist physicist, to philosophize in terms of rela- 
tivity, is to be gathered from a very interesting letter addressed 
to us by a most distinguished physicist. 2 Inasmuch as other 
readers may have encountered the same difficulty and as none, 
surely, will have formulated it more clearly, we are going to 
quote the main points in this letter. We shall then reproduce 
our reply. 

Let AB be the trajectory of the projectile plotted in the system 
earth. Starting from point A on the earth, where Peter will remain, 
the projectile carrying Paul heads toward B at speed v; having 
arrived at B, the projectile turns around and heads back to point 
A at speed - v. Peter and Paul meet again, compare measurements, 
and exchange impressions. I say that they are not in agreement 
about the duration of the journey: if Peter asserts that Paul has 
stayed away a given length of time, which he has estimated at A, 
Paul will reply that he is quite sure he has not spent that much time 
on the trip, because he has himself calculated its duration with a 
unit of time defined in the same way and has found it shorter. Both 
will be right. .... . 

I am assuming that the trajectory has been staked out with identi- 
cal clocks, borne along with the earth, hence belonging to the system 
earth, and that they have been synchronized by light signals. In toe 
course of his journey, Paul can read the time shown by the particu- 
lar clock near which he is passing, and can compare this time witn 
that indicated by an identical clock in his projectile. _ 

You can already see how I am orienting the question: the point 
is to compare adjacent events, to observe a simultaneity of c lock 
readings at the same place. We are not straying from the psychologi- 
cal conception of simultaneity, for, in accord with your own expres- 
sion, an event E occurring beside clock C is given m 
with a reading on clock C in the psychologist's sense of the word 

'"Sent "departure of the projectile," 
clocks both point to 0'. I am assuming, of course, that thepjj^ 
attains its speed instantaneously. There, then, is the projecule th* 
constitutes a system S' traveling in rectilinear and uniform motion 

2[Bergson tactfully refrains from naming this physicist but ^ js iden- 
tinea BecquLl (1878-1953) by And* » ^ ^ 

stein et la nouvelle edition de l'ouvrage de M. Bergson uu 
taniiter Revue de philosophie, XXXI (1924), 241-260.] 



with respect to the system earth, at speed v. For the sake of clarity, 

I shall assume that v = 259,807 km/sec, so that the factor -v / 1 - 

equals . 

I shall assume that at the end of an hour, recorded on the clock 
of the projectile, the latter passes the middle M of the distance AB. 
Paul reads the time both on his clock (l c ) and, simultaneously, on 
the system earth's clock located at M. What time will he read on the 
latter? One of the Lorentz equations supplies the answer. 

We know that the Lorentz formulae give the relations linking the 
space and time co-ordinates of an event measured by Peter with the 
space and time co-ordinates of the same event measured by Paul. In 
the present case, the event is the meeting of the projectile with the 
system earth's clock at M; its co-ordinates in the projectile system S' 

1 / vx'\ 

are x' = 0, V = 1 °; the formula t = — — = I V + J gives t = 2 1' (since 

Paul therefore notes that the system earth's clock before which he 
is passing is one hour ahead of his; of course, he does not have to 
push his clock ahead; he records the disagreement. Continuing on 
his journey, he notes that the time differences between his clock and 
those he successively encounters increase in such proportion to his 
own clock-time that, on arriving at B, his clock points to 2 C ; but the 
system earth's clock at B points to 4 e . 

Having arrived at B, the projectile turns back along BA at speed 
-v. Now there is a change in system of reference. Paul abruptly 
leaves the system moving with speed +v with respect to the earth 
and passes into the system of speed -v. Everything starts over 
again on the return trip. Let us imagine that the clock in the pro- 
jectile and the one at B are automatically moved back to zero, and 
that the other earth-linked clocks are synchronized with the one at B. 
We can begin the preceding argument all over again: at the end of 
one hour's journey, recorded on Paul's clock, he will again find as 
he passes M that his clock reads 1°, whereas the earth clock reads 
2 C , etc. 

But why imagine the clocks set back to zero? It was useless to 
interfere with them. We know there is an initial shifting from zero 
to take into account; this shifting amounts to 2 C for the projectile's 
clock and 4" for the system earth's clock; they are constants to be 



= 2). The clock at point M therefore records 2 C . 



added to the times that would be shown had all the clocks been 
pushed back to zero. Thus, if we have not interfered with the clocks, 
when the projectile recrosses M, Paul's clock will show 1+2 = 3°, the 
one at point M, 2 + 4=6°, and Peter's 4 + 4 = 8°. 

Behold the result! For Peter, who has remained at A on the earth, 
it is indeed eight hours that have elapsed between Paul's departure 
and return. But, if we ask "living, conscious" Paul, he will say that 
his clock read 0° at departure and reads 4° upon return, that it has 
recorded a duration of 4°, and that he has really been traveling 4° 
and not 8 C . 

So goes the objection. As we stated, it is impossible to pre- 
sent it in clearer terms. That is why we have reproduced it 
just as it was addressed to us, without reformulation. Here, 
then is our reply: 

"Two important remarks must be made at the outset. 

1. If we take a stand outside the theory of relativity, we 
conceive of absolute motion and, therewith, absolute immo- 
bility; there will be really motionless systems in the universe. 
But, if we assume that all motion is relative, what becomes of 
immobility? It will be the state of the system of reference, the 
system in which the physicist imagines himself located, inside 
which he is seen taking measurements and to which he relates 
every point in the universe. One cannot move with respect to 
oneself; and, consequently, the physicist-builder of Science, is 
motionless by definition, once the theory of relativity is ac- 
cepted. It unquestionably occurs to the relativist physicist, as 
to any other physicist, to set in motion the system of reference 
in which he had at first installed himself; but then, willy-nilly, 
consciously or unconsciously, he adopts another, if only for an 
instant; he locates his real personality within this new system, 
which thus becomes motionless by definition; and it is then no 
more than an image of himself that he mentally perceives m 
what was just now, in what will in a moment again become, 
his system of reference. 

2. If we stand outside the theory of relativity, we can quite 
readily conceive of an absolutely motionless individual, Peter, 
at point A, next to an absolutely motionless cannon; we 
can also conceive of an individual, Paul, inside a projectile 



launched far out from Peter, moving in a straight line with 
absolutely uniform motion toward point B and then return- 
ing, still in a straight line with absolutely uniform motion, to 
point A. But, from the standpoint of the theory of relativity, 
there is no longer any absolute motion or absolute immo- 
bility. The first of the two phases just mentioned then becomes 
simply an increasing distance apart between Peter and Paul; 
and the second, a decreasing one. We can therefore say, at will, 
that Paul is moving away from and then drawing closer to 
Peter, or that Peter is moving away from and then drawing 
closer to Paul. If I am with Peter, who then chooses himself as 
system of reference, it is Peter who is motionless; and I explain 
the gradual widening of the gap by saying that the projectile 
is leaving the cannon, and the gradual narrowing, by saying 
that the projectile is returning to it. If I am with Paul, now 
adopting himself as system of reference, I explain the widen- 
ing and narrowing by saying that it is Peter, together with the 
cannon and the earth, who is leaving and then returning to 
Paul. The symmetry is perfect. 3 We are dealing, in short, with 
two systems, S and S', which nothing prevents us from assum- 
ing to be identical; and one sees that since Peter and Paul re- 
gard themselves, each respectively, as a system of reference and 
are thereby immobilized, their situations are interchangeable. 
I come now to the essential point. 

If we stand outside the theory of relativity, there is no ob- 
jection to expressing ourselves like anyone else, to saying that 
both Peter and Paul, the one absolutely motionless and the 
other absolutely in motion, exist at the same time as conscious 
beings, even physicists. But, from the standpoint of the theory 
of relativity, immobility is of our decreeing: that system be- 
comes immobile which we enter mentally. A "living, con- 
scious" physicist then exists in it by hypothesis. In short, Peter 

3 It is perfect, we repeat, between Peter and Paul as the referrers, as it is 
between Peter and Paul as the referents. Paul's turning back has nothing 
to do with the matter, since Peter turns back as well if Paul is the re- 
ferrer. We shall, moreover, directly demonstrate the reciprocity of accelera- 
tion in the next two appendixes. 


is a physicist, a living, conscious being. But what of Paul? If I 
leave him living and conscious, all the more if I make him a 
physicist like Peter, I thereupon imagine him taking himself 
as system of reference, I immobilize him. But Peter and Paul 
cannot both be motionless at one and the same time, since, by 
hypothesis, there is first a steadily increasing and then a stead- 
ily decreasing distance between them. I must therefore choose 
between them; and, in point of fact, I did choose, since I said 
that it was Paul who was shot into space and thereby immobi- 
lized Peter's system into a system of reference. 4 But then, Paul 
is clearly a living, conscious being at the moment of leaving 
Peter; he is still clearly a living, conscious being at the moment 
of returning to Peter (he would even remain a living, conscious 
being in the interval if, during this interval, we agreed to lay 
aside all questions of measurement and, more especially, all 
relativist physics); but, for Peter the physicist, making measure- 
ments and reasoning about them, accepting the laws of physico- 
mathematical perspective, Paul, once launched into space, is 
no more than a mental view, an image-what I have called 
a "phantom" or, again, an "empty puppet." It is this Paul 
en route (neither conscious nor living, reduced to the state ot 
an image) who exists in a slower time than Peter's. It would 
therefore be useless for Peter, attached to the motionless system 
that we call earth, to try to question this particular Paul at 
the moment of his re-entering the system, about his travel 
impressions: this Paul has noted nothing and had no impres- 
sions, since he exists only in Peter's mind. What is more, ne 
vanishes the moment he touches Peter's system. The Paul wn 
has impressions is a Paul who has lived in the interval, ana 
the Paul who has lived in the interval is a Paul who was inter- 
changeable with Peter at every moment, who occupied a 

* It is clearly by extension that use has been made of ^ e e ^ZTZ 
"system of reference" in the passage from the above-quoted letter, ^ 
which it was stated that Paul, in turning back, "changes n y 
reference." Paul is really, by turns, in systems ^^^JS in 
reference; but neither of these ™ ^ J^w footnote 4 

motion, is a system of reference. See Appendix III, particular y 

on pp. 184-185. 



identical with Peter's and aged just as much as Peter. Every- 
thing the physicist will tell us about Paul's findings on his 
journey will have to be understood as being about findings 
that the physicist Peter attributes to Paul when he makes him- 
self a referrer and considers Paul no more than a referent- 
findings that Peter is obliged to attribute to Paul as soon as he 
seeks a picture of the world that is independent of any system 
of reference. The Paul who gets out of the projectile on re- 
turning from his journey and then again becomes part of 
Peter's system, is something like a flesh-and-blood person step- 
ping out of the canvas upon which he had been painted: it 
was to the portrait, not the person, to Paul referent, not re- 
ferrer, that Peter's arguments and calculations applied while 
Paul was on his journey. The person replaces the portrait, 
Paul referent again becomes Paul referrer or capable of refer- 
ring, the moment he passes from motion to immobility. 

But I must go into more detail, as you yourself have done. 
You imagine the projectile impelled by speed v such that we 
I ^ i 

have yl 1 - — = . Let AB then be the trajectory of the pro- 
jectile plotted in the system earth, and M the middle of the 
straight line AB. "I shall assume," you say, "that at the end 
of an hour recorded on the clock in the projectile, the latter 
passes the middle M of the distance AB. Paul reads the time 
both on his clock (P) and, simultaneously, on the system 
earth's clock located at M. What time will he read on the 
latter, if both clocks pointed to 0 C at departure? One of the 
Lorentz equations gives the answer: the clock at M points 
to 2V 

I reply: Paul is incapable of reading anything at all; for, 
insofar as, according to you, he is in motion with respect to 
motionless Peter, whom you have made referrer, he is nothing 
more than a blank image, a mental view. Peter alone will 
henceforth have to be treated as a real, conscious being (unless 
you renounce the physicist's standpoint, which here is one of 
measurement, to return to the standpoint of common sense or 


ordinary perception). Hence we must not say, "Paul reads the 
time. . . ." We must say, "Peter, that is, the physicist, pictures 
Paul reading the time. . . ." And, since Peter applies, and must 
apply, the Lorentz equations, he naturally pictures Paul read- 
ing P on his moving clock at the moment when, in Peter's 
view, this clock passes in front of the clock of the motionless 
system, which, in Peter's eyes, points to 1°. But, you will tell 
me: "Nonetheless, does there not exist in the moving system, 
a moving clock that records its own particular time independ- 
ently of anything Peter can imagine of it?" Without any doubt. 
The time of this real clock is exactly what Paul would read 
on it if he became real again, I mean, alive and conscious. But, 
at this precise moment, Paul would become the physicist; he 
would take his system as the system of reference and immobi- 
lize it. His clock would then point to 2-exactly the time to 
which Peter's clock pointed. I use the past tense because al- 
ready Peter's clock no longer points to V but to 1", being now 
the clock of Peter referent and no longer referrer. 

I need not pursue the argument. Everything you said about 
the times read by Paul on his clock when he arrives at B, then 
when he comes back to M, and, finally, when he is about to 
touch A and re-enter the system earth, all this applies not to 
living, conscious Paul, actually looking at his moving clock, 
but to a Paul whom physicist Peter pictures as watching this 
clock (and whom the physicist must picture in this way and 
need not distinguish from a living, conscious Paul: this dis- 
tinction is the philosopher's concern). It is for this merely 
imagined and referred-to Paul that four-imagined-hours will 
have elapsed while eight-lived-hours will have elapsed tor 
Peter. But Paul, conscious and therefore referrer, will nave 
lived eight hours, since we shall have to apply to him every- 
thing we just said about Peter." 

To sum up, in this reply we once more gave the meaning 
of the Lorentz equations. We have described this ™ eanin S m 
many ways; we have sought by many means to present a 
crete vision of it. One could just as easily have established 



in abstracto in the standard step-by-step deduction of these 
equations. 5 One would recognize that the Lorentz equations 
quite clearly express what the measurements attributed to S' 
must be in order that the physicist in S may see the physicist 
imagined by him in S', finding the same speed for light as he 

5 Albert Einstein, La theorie de la relativity restreinte et generalisee, pp. 
101-107; Jean Becquerel, Le principe de relativity et la theorie de la gravi- 
tation, pp. 29-33. 


The Reciprocity of Acceleration 

In the preceding Appendix, as in our fourth chapter, we 
broke down the journey in the projectile into two journeys m 
opposite directions, both of which were uniform translations. 
There was no point in bringing up the difficult that attach 
or seem to attach, to the idea of acceleration: in the course ot 
this work, we have never declared for reciprocity anywhere 
except in the case of uniform motion, where it is obvious But 
we could just as well have taken into account the a«dm*» 
that the change of direction gives rise to and then have con 
sidered the entire journey in the projectile as a variable mo 
tion. Our argument would have held, for we shah , « t thX 
acceleration is itself reciprocal and that the two systems S 
S' are entirely interchangeable. „ rr plera- 

One sometimes hesitates to admit this reciprocity of ac el * 
tion for certain special reasons, which will concer m» n «i § 
next Appendix, when we shall be dealing with Wcridto*_ 
But onealso hesitates because, as it is usually s a, ,d a«e * 
ated motion in a moving system is conveyed 
that do not occur symmetrically m the s y stei * fo deal . 
less, which has been taken as the system of reteren . . 

„ trarV one agrees to spcai* 
ing with a train moving on a tracK, 01 & ^ trans . 
reciprocity as long as the motion rema, ris umto . ^ ^ 
lation, it is thought, can be attributed equally t ^ 

to the train; all that the r^J*§£ u*«**>« 
asserts about the moving train cou d as w 
the track, which has become mobile by the pi y ^ ^ 

onto the train. But let the speed of the trai ^ ^ ^ 
crease abruptly, let it stop: the physicist HencCj 
a jolt, and this jolt has no counterpart on the 



no more reciprocity in the case of acceleration; the latter mani- 
fests itself in phenomena at least some of which concern only 
one of the two systems. 

There is a grave confusion here, whose causes and effects 
it would be interesting to probe. Let us limit ourselves to 
describing its nature. One continues to see a single system in 
what has just been revealed as a collection of systems, a mani- 
fold of different systems. 

To be immediately persuaded of this, we have only to render 
the two systems under consideration actually indecomposable 
by making, say, two physical points out of them. It is clear that 
if point S' is in variable rectilinear motion with respect to S 
ruled motionless, 5 will have a variable rectilinear motion of 
the same speed at the same moment with respect to S' ruled 
motionless in its turn. 1 But we can just as readily attribute to 
S and S' any dimension and any motion of translation we like: 
if we adhere to our hypothesis, namely, that each of the two 
is and remains a system, that is, a group of points compelled 
constantly to keep the same relative positions with respect to 
one another, and if we agree to consider only translations, 2 it is 
obvious that we shall be able to treat them as if they were two 
physical points, and that their acceleration will be reciprocal. 

To these systems S and S' in any state of reciprocal transla- 
tion whatever, there will moreover apply, as far as time is 
concerned, everything we said about reciprocal motion when 
it was uniform. Let S be the system of reference: 5' will have 
changing speeds, each of which will be kept up for finite or 

Ut would be inaccurate, moreover, to say that these speeds are in oppo- 
site directions. To attribute speed in opposite directions to two systems 
would consist, at bottom, of mentally settling in a third system of refer- 
ence, when we have given ourselves only S and S'. Let us rather say that 
the direction of speed will have to be described in the same way in both 
cases because whether we adopt S as system of reference or whether we 
prefer taking our place in S', in both cases the motion we attribute from 
there to the other system is a motion that brings the mobile nearer or 
sends it farther away. In a word, the two systems are interchangeable and 
whatever we say in S about 5' can be repeated in S' about S. 
2 The case of rotation will be examined in the next Appendix. 


infinitely short periods; to each of these motions the Lorentz 
formulae will, of course, apply; and we shall obtain, either by 
an addition of finite parts or by an integration of infinitely 
small elements, the time t' which is judged to elapse in S' while 
time i is elapsing in S. Here again, V will be smaller than f; 
here again, there will have been an expansion of the second 
and a slowing of time as a result of motion. But here again 
the shorter time will be merely attributed time, incapable of 
being lived, unreal: only, the time of S will be a time that 
could be lived, a time that is, moreover, actually so lived, a 
real time. Now, if we take S' as our system of reference, it ism 
S' that this same real time will elapse and into S that the 
imaginary time t' will be transferred. In a word, if there is 
reciprocity in the case of accelerated motion, as in that of uni- 
form motion, the slowing of time for the system assumed in 
motion will be figured the same way in both cases, a slowing 
again only imagined and not affecting real time. 

The symmetry between S and S' is therefore perfect, inso- 
far as S and S' are really two systems. 

But, without noticing it, we sometimes substitute tor ne 
system ruled in motion a number of separate systems endowed 
with different motions, which we nevertheless continue to trea 
as a single system. We often do this even when we speak o 
phenomena "inside the system" which occur as the result jt 
this system's accelerated motion and when for examp , 
are shown a passenger jolted in his se,by^* 
stop. If the passenger is shaken up, it is cieany 
Physical points of which his body is composed do not^main 
tain unchanging positions with ^P**^^ do 'not 
general, with respect to one another, incy ^ 
form a single system with the train or fay ^ 

selves-as many systems S , and S , etc Consequently, 
"jolt" as are endowed with motions of their own ^ 
in the eyes of the physicist in 5, they have tn 
f", etc. The reciprocity is, moreover, still co F we 
S and S", and between S and S», as between Sand 
install the real physicist, by turns, in i , >> > 



be in several at the same time), he will find and live the same 
real time t in each, in that event successively attributing the 
merely conceived times t", V", etc., to system S. This means 
that the passenger's jolt introduces no asymmetry. 3 From the 
standpoint we have to assume, it is dissolved into perfectly 
reciprocal manifestations affecting the invariable point-systems 
with which we are dealing. The standpoint we must assume 
is, in fact, that of the measurement of time in the theory of 
relativity, and the clocks of which this theory speaks can 
clearly be likened to ordinary physical points, since their sizes 
are never taken into account. It is, therefore, really ordi- 
nary physical points that are in motion, in the case of ac- 
celerated as in that of uniform motion, when we compare 
the times of these reciprocally moving clocks in the theory of 
relativity. In short, it matters little whether the motion is uni- 
form or variable: there will always be reciprocity between the 
two systems that we bring face to face. 

This, moreover, is what we are about to see with more pre- 
cision in the next Appendix, where we shall consider the 
reciprocity of acceleration in all its generality. The points Af x 
and M 2 with which we shall first deal can be considered clocks 
as well. 

3 Here, as elsewhere, we must remember that science retains, and must 
retain, only the visual aspect of motion. The theory of relativity requires 
before all, as we have shown (pp. 32ff), that we apply this principle with 
utmost rigor. We sometimes forget this when we speak of the jolt felt by 
our passenger. Whoever wishes to think in terms of relativity must begin 
by either eliminating the tactile or transposing it into the visual. If we 
resolve the jolt into its visual elements, and if we keep in mind the mean- 
ing of the word "system," the reciprocity of acceleration again becomes 
apparent. We must, moreover, guard against the temptation mentally to 
enter systems S", S'", etc., at the same time. We do this when we speak 
of the jolt-even reduced to what we see of it, as of a single fact. We must, 
indeed, distinguish between the point of view of perception and that of 
science. Perception undoubtedly embraces S", S'", etc., all at one time. But 
the physicist cannot adopt them in the ensemble as a system of reference: 
he must select one of them, considering them one at a time. 


"Proper-Time" and "World-Line" 

We have just demonstrated the reciprocity o£ acceleration, 
first in a particular case, then in a more general way. It is 
natural for this reciprocity to escape our attention when the 
theory of relativity is presented in its mathematical form, we 
implied the reason for this in our sixth chapter,* where we 
stated (1) that the theory of relativity is obliged to rank the 
"real vision" with the "virtual vision," the measurement actu- 
ally made by an existing physicist with the one considered 
made by a merely imagined physicist; (2) that the form given 
this theory since Minkowski has precisely the effect ot hiding 
the difference between the real and the virtual, between what 
is perceived, or perceptible, and what is not. The reciprocity 
of acceleration appears only if we restore this distinction sec- 
ondary for the physicist, fundamental for the philosopher At 
the same time the meaning of the "slowing" that acce lerauon 
imparts to a moving clock is realized. It is realized without 
there being anything to add to what we said when tre ^^ 
uniform motion: acceleration cannot create new 00 
here, since one must still apply the Lorentz formulae s(m g ;en 
eral, to infinitesimal elements) when one speaks <* ^*F£ 
slowed times. But, for greater precision, we are gm 8 
amine in detail the special form which the theory of relatm y 
exhibits in this case. We take it from a recent book that i^ 
already a classic, the important work of Jean Becq v 
Principe de relativite et la theorie de la gravitation [Fans. 
Gauthier-Villars Cie, 1922], pp. 48-51). 

In a system of reference connected with a portion of matter, 

1 Particularly pp. 131ff., and pp. 152ff. 




is, in a system all of whose points are in the same state of motion 
any motion, as this portion of matter, the spatial distance between 
two events relating to this portion of matter is always zero. We there- 
fore have, in this system in which dx = dy = dz = 0, 

ds = cdr, I ds=c I dr, 

a ■ , Ja Ja ' 

dr ,s the proper-time element of the portion of matter considered 
and of the whole system connected with it. The proper-time f B dr 

elapsed between two events A and B is the time an observer will 
compute the time that the clocks in the system will record. 

A clock attached to a mobile (whose motion need now no longer 
be subject to the restriction of uniform translation) computes the 
length, divided by c, of the arc of the World-line of this mobile. 

Let us now consider a free physical point M v Galileo's law of 
nertia informs us that this point is in rectilinear, uniform motion: 
to tim state of motion there corresponds, in space-time, a "World- 
line formed by the block of events that represent the different, 

ZTZZ 6 P°, SItl ° nS ° f , this m ° bil * * its state of uniform motion 
positions that we can plot in any system at all. 

AnM Wo J d lille °f I« us pick out two determinate events 
ber of ™i W ^ eVentS WC C3n ima & ine an infi ™te num- 

moMe aT w h V t0 , d ° WC nCed 0n] y contemplate a second 
"oner CVent A and ^versing a longer or 

hall diK dlstance .at a greater or lesser speed, a distance we 

Wjs;;^ translation connected with M - 

are^nToSe 0 "' ^ ^ f ° ll0WS: the tWO mobiles M i and M * 

Slv we t „^ : Ml 18 385,11116(1 in uniform translation. 

Jt is 'imDoS^ T" tS V 8y8tem * connected with M v 

S a A to to ■ ! th3t Ms ' haVin ^ left the "niforrn system 

saJ y under^re an f ( " ^ to p3SS OUt of h at *>• "as neces- 
sarily undergone an acceleration between events A and B 

be^i^T.ft? ? timC ' 3nd < + d < in ^-^ -eluded 
SSS^ M, At the ^oTntslrf 11 ^ * *™» " ^ S 

mobile Af 2 is referenced 7/??^ V ^ ' + d< ' the SCC ° nd 
system S; 'these co-ordtat'es" locate o n*£ftf + ift- Z + ^ < + ^ * 
infinitely adjacent events C and n I World " llne of M 2. two 
/j events C and D, whose interval is ds; we have 2 

than™ 6 r~V d o;rr; h iS m ° St ° ftCn *" ™? ma- 

nner adopted in the present work), in order to keep S> 

"proper-time" and "world-line" 


ds 2 = - dx 2 - dy 2 - dz 2 + c 2 dt 2 . But we also have ds = cdr, dr being the 
element of proper-time of the mobile M 2 . From this, we deduce 3 

ds 2 = c 2 dr 2 = c 2 dt 2 1 ■ 

c 2 [\dt J + \dt) + \dt) 

: C 2 dt 2 ^1 -^j = a 2 C 2 dt 2 

v being the speed of the mobile M 2 at the point of time t, both 
speed and time being computed in the uniform system of mobile M v 
We therefore finally have 

0) dr=adt, 

which means: the proper-time of a mobile M 2 between two events 
on its World-line is shorter than the time computed between the 
same events in a system in uniform translation; it is as much shorter 
as the speed of the mobile with respect to the uniform system is 
greater. . . . 

We have not yet taken note of the absolute coincidence of mobiles 
Af x (in uniform translation) and M 2 (any motion), at events A and B. 
Let us integrate (1) 

J A Jt A 


the more the motion of the mobile M 2 between events A and B 
common to the two moving points differs from a rectilinear, uniform 
motion, the greater will be its speeds with respect to M v since the 
total duration t B - t A is fixed, and the shorter the total proper-time 
will be. 

In other words: between two determinate events, the longer 
World-line is the one corresponding to the motion of uniform trans- 

[It is important to observe that, in the preceding demonstration, 
here is no reciprocity between the systems of reference connected 
w «h M l and M 2 , because M 2 is not in uniform translation. The 
acceleration of M 2 has created the asymmetry: here one recognizes 
the absolute character of acceleration.] 

from being negative, as would happen in the most frequent case, that in 
ich the distances between two events in space is shorter than the path 
raversed by light during the interval of time that separates them. This 
is the only one in which, according to the theory of relativity, one of 
6 tWo eve "ts can act upon the other. This is precisely the hypothesis 
wat is assumed above. 

8 The factor |/l -- ; s here designated by a. 



Strange consequences follow from the results just established. 

In a system in uniform translation— the earth, for example, be- 
cause its acceleration is slight— two identical, synchronized clocks are 
at the same spot. We shift one very rapidly and bring it back again 
close to the other at the end of time t (the time of the system); it is 

found to be behind the other clock by f- adt; if its accelera- 
tion was instantaneous at departure as upon arrival and its speed 
has remained constant, the slowing amounts to t(l-a). 

No one could express himself with greater precision. More- 
over, from the physico-mathematical standpoint, the argument 
is irreproachable: the physicist ranks the measurements actu- 
ally made in one system with those which, from this system, 
appear as if actually made in another. It is out of these two 
kinds of measurement, merged in the same treatment, that he 
constructs a scientific world-view; and, as he must treat them 
in the same way, he gives them the same meaning. Quite dif- 
ferent is the philosopher's role. In a general way, he wants to 
distinguish the real from the symbolic; more exactly and more 
particularly, for him, the question here is to determine which 
is the time lived or capable of being lived, the time actually 
computed, and which is the time merely imagined, the time 
which would vanish at the very instant that a flesh-and-blood 
observer would betake himself to the spot in order to compute 
it in actuality. From this new point of view, comparing only 
the real with the real, or else, the imagined with the imagined, 
we see complete reciprocity reappearing, there where accelera- 
tion seemed to have brought on asymmetry. But let us closely 
examine the text we just quoted. 

We notice that the system of reference is defined there as 
"a system all of whose points are in the same state of motion." 
The fact is that the "system of reference connected with M" 
is assumed in uniform motion, while the "system of reference 
connected with M 2 " is in a state of variable motion. Let S and 
S' be these two systems. It is clear that the real physicist then 
gives himself a third system S" in which he imagines himself 

"proper-time" and "world-line" 


installed and which is thereby immobilized; only with respect 
to this system can S and S' be in motion. If there were only 
S and S', he would necessarily place himself in S or in S', and 
necessarily one of the two systems would be found immobi- 
lized. But, the real physicist being in S", the real time, that is, 
the lived and actually measured time, is the one in system S". 
The time of system S, being the time of a system in motion 
with respect to S", now becomes a slowed time; it is, more- 
over, only an imagined time, that is, attributed to system S by 
the observer in S". In this S system an observer has been imag- 
ined who takes it as his system of reference. But, once again, 
if the physicist really took this system as his system of refer- 
ence, he would be placing himself within it, he would be 
immobilizing it; since he remains in S" and leaves system S in 
motion, he is limited to picturing an observer taking S as sys- 
tem of reference. In short, we have in S what we called a 
phantom observer, judged to be taking as his system of refer- 
ence this S system that the real physicist in S" pictures in 

Moreover, between the observer in S (if he became real) and 
the real observer in S" the reciprocity is perfect. The phan- 
tom observer in S, turned real again, would immediately re- 
discover the real time of system 5", since his system would be 
immobilized, since the real physicist would have transported 
himself to it, since the two systems, as referrers, are inter- 
changeable. The phantasmal time would now be elapsing in 

Now, everything we just said about S with respect to S" we 
can repeat for system S' with respect to this same S" system. 
Real time, lived and actually computed by the physicist in S" 
will again be present in motionless S". This physicist, taking 
hl s system as system of reference will attribute to S' a slowed 
^me, one which is now of variable rhythm, since the speed of 
system varies. Moreover, at each instant, there will again 
°e reciprocity between S" and S'; if the observer in S" were to 
transport himself into S', the latter would at once be immobi- 



lized and all the accelerations that were present in S' would 
pass into S"; the slowed, merely attributed times would pass 
with them into S", and it is in S' that time would be real. 

We have just considered the relation of motionless S" to S 
in uniform translation, then the relation of motionless S" to 
5' in a state of variable motion. There is complete reciprocity 
in both cases— provided we consider both the systems we are 
comparing as either referrers, when entering them one at a 
time, or as referents when leaving them one at a time. In both 
cases there is a single, real time, the one which the real physi- 
cist first noted in S" and finds again in S and S' when he trans- 
ports himself into them, since S and 5" are interchangeable as 
referrers, as are also S' and S". 

It remains then to consider directly the relation of S in uni- 
form translation to S' in variable motion. Now we know that 
if S is in motion, the physicist who is found in it is a merely 
imagined physicist-the real physicist is in S". The system of 
reference really adopted is S", and the system S is not a real 
system of reference but an imagined system of reference that a 
merely imagined observer adopts. This observer is already 
phantasmal. Doubly phantasmal then is his noting of what is 
happening in S'; it is a mental view attributed to an observer 
who is himself only a mental view. Thus, when it is stated, in 
the above-mentioned text, that there is asymmetry between S 
and S', it is clear that this asymmetry does not concern meas- 
urements really taken in either S or S', but those which are 
attributed to the observer in S from the standpoint of S", and 
those which, still from the standpoint of S", are considered to 
be attributed by the observer in S to the observer in S'. But, 
in that case, what is the true relation between the real S and 
the real S'? 

To discover it, we have only to place our real observer in S 
and S' by turns. Our two systems will thus become successively 
real, but also successively motionless. We could, moreover, 
have taken this path right away without passing through such 
a long detour, by following the quoted text to the letter and 
considering only the special case in which system S, which we 

"proper-time" and "world-line" 


are told is in uniform motion, moves at a constant speed of 
zero. Here, then, is our real observer in S, now motionless. It is 
clear that this observer in S will discover that there is no reci- 
procity between his own motionless system and system S' which 
leaves it to rejoin it later. But, if we place him now in S', 
which will thus be found immobilized, he will note that the 
relation of S to S' is just what the relation of S' to S was a mo- 
ment ago: it is now S which leaves S' and which has just 
rejoined it. Thus, there is symmetry once again, complete reci- 
procity between S and S', referrer, and S' and S, referent. 
Acceleration therefore changes nothing in the situation; in the 
case of variable motion, as in that of uniform motion, the 
rhythm of time varies from one system to another only if one 
of the two systems is referrer and the other, referent, that is, 
if one of the two times is capable of being lived, is actually 
computed, is real, while the other is incapable of being lived, 
is merely conceived as computed, is unreal. In the case of vari- 
able motion as in that of uniform motion, asymmetry exists 
not between the two systems but between one of the systems 
and a mental view of the other. It is true that the quoted text 
clearly shows us the impossibility of mathematically expressing 
this distinction in the theory of relativity. The consideration 
of World-lines" introduced by Minkowski even has as its 
essence the masking or rather the wiping out of the differ- 
ence between the real and the imagined. An expression like 
ds 2 = ~ dx 2 - dy 2 - dz 2 + c 2 dt 2 seems to place us outside every 
system of reference, in the Absolute, in the presence of an 
entity comparable to the Platonic Idea. Then, when we apply 
it to specific systems of reference, we think we are particular- 
lz mg and materializing an immaterial, universal essence, as the 
latonist does when he descends from the pure Idea, contain- 
in g immanently all the individuals of a genus, to any one 
among them. All systems then acquire equal rank; all assume 
the same value; the one in which we have dx = dy = dz = 0 be- 
comes just another system. We forget that this system harbored 
e rea 1 Physicist, that the others are only those of imagined 
P ysicists, that we had been looking for a mode of representa- 



tion suitable to the latter and the former at the same time, 
and that the expression ds 2 - - dx 2 - dy 2 - dz 2 + c 2 dt 2 had been 
precisely the result of that search. It is therefore truly begging 
the question to hold up this general expression as authority 
for equating every system and declaring all times of equal 
worth, since this community of expression was obtained only 
by neglecting the difference between the time in one of them— 
the only verified or verifiable, the only real time— and the 
merely imagined, fictional times in all the others. The physi- 
cist had the right to wipe out this difference. But the philoso- 
pher must re-establish it. This is what we have done. 4 

* In a word, the theory of relativity requires that the physicist be in- 
stalled in one of the systems he gives himself, in order to assign from 
there a particular motion to each of the other systems, since there is no 
absolute motion. He can choose any one of the systems in his universe; 
he can, moreover, change systems at any moment; but he is obliged to be 
in one of them at a particular moment. As soon as he clearly realizes this, 
the reciprocity of acceleration becomes clear to him, for the system in 
which he installs himself is interchangeable with any other system he is 
considering, whatever its motion, provided this system is conceived in itself 
and not in the perspective representation in which he provisionally sees it. 
Moreover, real time is what the physicist perceives and measures, what 
exists in the system in which he is installed; precisely because the moving 
system considered by him would be, when at rest, interchangeable with 
his at rest, our physicist would rediscover this same real time in the 
moving system being considered were he to project himself into it and, 
by that very fact, immobilize it, driving out then the phantasmal time 
which he had imagined in it and which, in actuality, could not be directly 
measured by anyone. But, precisely because he can imagine himself any- 
where and shift at each instant, he likes to picture himself everywhere or 
nowhere. And, as all systems no longer then appear to him as referred to 
one among them-his own-all pass onto the same plane: in all of them at 
once he thus installs physicists who would be kept busy referring even 
though, alone motionless for the moment, our physicist is really the only 
referrer. This, at bottom, is what he is doing when he speaks of "systems 
of reference in motion." Each of these systems can undoubtedly become 
a system of reference for the physicist actually referred to, who will be- 
come a referrer, but it will then be motionless. As long as our physicist 
leaves it in motion, as long as he regards all these purely mental construc- 
tions simply as possible systems of reference, the only true system of refer- 
ence is system 5" in which he himself has settled, in which he really 
computes time, and from which he then imagines those systems in motion 

"proper-time" and "world-line" 


In short, there is nothing to change in the mathematical 
expression of the theory of relativity. But physics would render 
a service to philosophy by giving up certain ways of speaking 
which lead the philosopher into error, and which risk fooling 
the physicist himself regarding the metaphysical implications 
of his views. For example, we are told above that "if two iden- 
tical, synchronized clocks are at the same spot in the system 
of reference, if we shift one very rapidly and then bring it 
back again next to the other at the end of time t (the time of 

the system), it will lag behind the other by t- adt." In 

reality we should say that the moving clock exhibits this slow- 
ing at the precise instant at which it touches, still moving, the 
motionless system and is about to re-enter it. But, immediately 
upon re-entering, it points to the same time as the other (it 
goes without saying that the two instants are practically indis- 
tinguishable). For the slowed time of the moving system is only 
attributed time; this merely attributed time is the time indi- 
cated by a clock hand moving before the gaze of a merely 
imagined physicist; the clock before which this physicist is 
situated is therefore only a phantasmal clock, substituted for 
the real clock throughout its journey: from phantasmal it again 
turns into real the moment it is returned to the motionless 
system. It would, moreover, have remained real for a real ob- 
server during the trip. It would not have undergone any slow- 

which are only potentially referrers. It is from the vantage of this system 
S" that he really operates-even if he mentally sees himself everywhere or 
nowhere-when he portions out the universe into systems endowed with 
this or that motion. The motions are such and such only with respect 
to S"; there is motion or immobility only with respect to S". If the physi- 
cist were really everywhere or nowhere, all these motions and immobili- 
zes w °uld be absolute ones; we would have to say goodbye to the theory 
°£ relativity. Relativity theoreticians sometimes seem to forget this; nor, 
a gain, is it anything of which they need take notice as physicists since, as 
We have shown, the distinction between the real and possible vision, be- 
tween the system of reference which is really adopted and the one merely 
"nagined as such, necessarily disappears in the mathematical expression 
of 016 theory. But the philosopher must re-establish it once more. 




ing. And that is precisely why it shows no slowing when it is 
again found to be a real clock upon arrival. 

It follows that our remarks apply equally to clocks placed 
and displaced in a gravitational field. 5 According to the theory 
of relativity, what is gravitational force for an observer in the 
system becomes inertia, motion, acceleration, for an observer 
outside of it. In that case, when we are told of "modifications 
undergone by a clock in a gravitational field," is it a question 
of a real clock perceived in the gravitational field by a real 
observer? Obviously not; in the eyes of the latter, gravitation 
signifies force, not motion. But it is motion, and motion alone, 
that slows the course of time according to the theory of rela- 
tivity, since this slowing can never be posited except as a conse- 
quence of the Lorentz formulae. 6 Hence, it is for the observer 
outside the field, mentally reconstructing the position of the 
clock hand but not seeing it, that the running of the clock is 
modified in the gravitational field. On the other hand, real 
time, indicated by the real clock, lived or capable of being 
lived, remains a time of unchanging rhythm; only a fictional 
time, which cannot be lived by anything or anyone, has its 
rhythm modified. 

Let us take a simple case, selected by Einstein himself, 7 that 
of a gravitational field created by the rotation of a disk. On a 
plane S adopted as system of reference and by that very fact 
immobilized, we shall consider a motionless point 0. On this 
plane we shall set a perfectly flat disk whose center we shall 
have coincide with point 0, and we shall have the disk turn 
about a fixed axis perpendicular to the plane at this point. We 

5 Insofar as these clocks would be affected by the intensity of the 
field. We are now leaving aside the consideration, with which we have 
been occupied till now, of the slowing that overtakes the clock by the 
mere fact of its leaving and returning to its position. 

6 And since it depends solely, as we have shown (pp. 117ff.), upon the 
lengthening of the "light-line" for the person who, outside the system, 
imagines the "light-figure" distorted as the result of its motion. 

™ l f inStdn ' ia thiorie de l ° relative restreinte et generalise pp. 
68-70. Cf. Jean Becquerel, Le principe de la relativiti et la thiorie de la 
gravitation, pp. 134-136. 

"proper-time" and "world-line" 


shall thus obtain a true gravitational field in the sense that an 
observer situated on the disk will note all the effects of a force 
pushing him away from the center or, as he will perhaps be- 
lieve, drawing him toward the periphery. It matters little that 
these effects do not follow the same law as those of natural 
gravitation, that they increase in proportion to the distance 
from the center, etc.: everything essential in gravitation is pres- 
ent, since we have an influence which, emanating from the 
center, is exerted upon objects standing out clearly on the disk, 
without taking into account the substance interposed, and 
produces on all things, whatever their nature or structure, an 
effect that depends only upon their mass and distance. Now, 
what was gravitation for the observer when he inhabited the 
disk, and thus immobilized it into a system of reference, will 
become an effect of rotational, that is, accelerated, motion 
when he betakes himself to point 0 of system S with which the 
center of the disk coincides, and when he gives this system, as 
we ourselves do, the status of a system of reference. If he pic- 
tures clocks located at various distances from the center of the 
disk's surface and considers them for a time short enough for 
their circular motion to be likened to a uniform transla- 
tion, he will, of course, believe that they cannot run synchro- 
nously, since their respective speeds are at that moment pro- 
portional to the distance separating them from the center: the 
Lorentz equations do indeed indicate that time slows down 
when speed increases. But what is this time which slows down? 
What are these nonsynchronous clocks? Are we dealing with 
the real time, with the real clocks perceived a moment ago by 
he real observer situated in what seemed to him to be a gravi- 
tational field? Obviously not. We are dealing with clocks that 
are P^tured in motion, and they can be pictured in motion 
on ty in the mind of an observer considered motionless in his 
tUrn ' that is, outside the system. 

°ne sees at what point the philosopher can be misled by a 
fanner of expression that has become current in the theory 
0 relativity. We are told that a physicist, setting out from 
P°«tt 0 with a clock and walking with it across the disk, would 



perceive, once he has returned to the center, that it is now 
slower than the clock, synchronized beforehand, which was left 
at point 0. But the clock that begins to slow down immediately 
upon setting out from point 0 is a clock which, from that 
moment on, has become phantasmal, being no longer the real 
clock of the real physicist-the latter has remained with his 
clock at point 0, detaching only a shadow of himself and of his 
clock onto the disk envisaged as moving (or else, each point of 
the disk, upon which he will actually settle, becoming, for that 
reason, motionless; his clock, having remained real, will every- 
where be motionless and everywhere work the same way). 
Wherever you put the real physicist, he will bring immobility 
with turn; and every point on the disk where the real physicist 
sits is a point from which the observed effect will have to be 
interpreted no longer in terms of inertia, but of gravitation; 
the latter, as gravitation, changes nothing in the rhythm of 
time or in the running of the clocks; it does so only when it is 
construed as motion by a physicist for whom the clocks and 
times of the system, where he no longer is,* have become 
mere mental views. Let us therefore say that if we keep our 

f °' hiS dOCk ' 3fter havin S trav eled t°warf the 
periphery of the disk, will return to 0 just as it was, running 
as before, not having slowed down. The theory of relativity 

prlciseZ/ ^ UireS thCre be a sIowin S down at 

112 T h T 11 r T ns °- But at that P recise ^ * 
pO^Z?" at the precise instant of leavir * the ^ 

the^vSstTn 0 ^' anal ° gOUS errOT ' admissibIe in 

h^^JZ 8 ™? ^ P hilos °P her > ^en we say that, 

fine Z ^ by meanstf Tot" "* * ™* ^ <° *» 
system" T, 1? u motionless with respect to the 

system. Is lt true that the disk constitutes a system? It is a 

8 When we say that the DhvsirUf ;„ , 
of course, that he does not wish to u n ° Ion « er ln the ^m, we mean, 
in the system; but he m^tl '*- m *? 7 ^ ^ ^ ^ Hw 
another as system of reTere^ce Z ' ° UtSide h and ^ 

terms of motion. moment he explains gravitation in 

"proper-time" and "world-line" 


system if we imagine it motionless; but we are then placing 
the real physicist upon it; and at any point on the disk where 
we have the real physicist with his real clock, there is, as we 
just saw, the same time. Time undergoes different slowings at 
different points on the disk; and clocks situated at these points 
cease to be synchronous, only in the imagination of the physi- 
cist who no longer adopts the disk and for whom the disk, 
being thus again found in motion, again comes under the 
Lorentz equations. But, in that case, the disk no longer consti- 
tutes a single system; it breaks up into an infinite number of 
separate systems. Let us actually track one of its radii, con- 
sidering the points at which this radius intersects the inside 
circumferences, infinite in number, which are concentric with 
that of the disk. These points are impelled at the same instant 
by different tangential speeds, the greater the speed, the far- 
ther from point 0: they therefore belong to different systems 
for the motionless observer at 0, who applies the Lorentz for- 
mulae; while a dt time elapses at 0, it is a slowed adt time that 
our observer will have to attribute to any one of these moving 
points, a depending, again, upon the speed of the mobile and, 
consequently, upon its distance from the center. Hence, con- 
trary to what is said, the "turning" field has a perfectly defin- 
able time when it constitutes a system, for then, bearing the 
physicist, it does not "turn"; this time is the real time to which 
all the system's real and therefore synchronous clocks actually 
point. It ceases to have a definable time only when it "turns," 
the physicist having transported himself to the motionless 
point 0. But, in that case, it is no longer one system, but an 
infinity of systems; and we shall naturally find on them an in- 
finity of times, all fictional, into which real time will have 
been pulverized, or, rather, evaporated. 
To sum up, we have a choice of one of two things. Either 
disk is considered as turning and gravitation is there re- 
solved into inertia: we are then viewing it from the outside; 
e living, conscious physicist does not dwell on it; the times 
at unwind on it are only conceived times; there will, of 
course, be an infinity of them; the disk will, moreover, not 



constitute a system or object, it will be the name we give to a 
collectivity; we shall obtain, for the application of the Lorentz 
formulae, as many separate systems as there are physical points 
impelled by different speeds. Or else, this same turning disk 
is considered motionless: its inertia of a moment ago becomes 
gravitation; the real physicist lives there; it really is a single 
system; the time we find on it is real, lived time. But, in that 
case, we find the same time on it everywhere. 

The Library of Liberal Arts 

Aeschylus, Prometheus Bound 
d'Alembert, J., Preliminary 
Discourse to the 
Encyclopedia of Diderot 
Aquinas, St. T., The Principles of 
Nature, On Being and 
Essence, On Free Choice, 
and On the Virtues in 

Amstotle, Nicomachean Ethics 
On Poetry and Music 
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ARNauld, A., The Art of Thinking 

(Port-Royal Logic) 
Augustine, St., On Christian 
On Free Choice of the Will 
J ac °n, P., The New Organon 
Jaron and Blau, eds., Judaism 
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Muse: Critical Essays of 
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crG ,io °p ,oMe,ai,hysic! 

L \9 > Principles, 
dialogues, and 

Principles of 

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Patriot King 
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Road to God 
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Bradley, F., Ethical Studies 
Burke, E., An Appeal from the 
New to the Old Whigs 
Reflections on the Revolution 

in France 
Selected Writings and 
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Butler, J., Five Sermons 
Calvin, J., On the Christian Faith 

On God and Political Duty 
Catullus, Odi et Amo: 

Complete Poetry 
Cicero, On the Commonwealth 
Cid, The Epic of the 

Croce, B., Guide to Aesthetics 
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Dewey, J., On Experience, Nature, 

and Freedom 
Diderot, D., Encyclopedia 

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Dryden, J., An Essay of Dramatic 

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Epictetus, The Enchiridion 
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Erasmus, D., Ten Colloquies 
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Feuerbach, L., Principles of the 

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Fichte, J., The Vocation of Man 
Goethe, J., Faust I and II (verse) 
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Faust II (prose) 
Grant, F., ed., Ancient Roman 
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Grottus, H., Prolegomena to The 
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Hamilton, C, ed., Buddhism 
Hanslick, E., The Beautiful in 

Hendel, C, Jean-Jacques 

Rousseau: Moralist 
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On History 
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Lao Tzu, The Way of Lao Tzu 
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Locke, J., A Letter Concerning 
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Mill, J., An Essay on Government 
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°valis, Hymns to the Night and 

Other Writings 
0c *HAM, W., Philosophical 

p Writings 

Pa?' «' Age 0f Reason 
p W - Natural Theology 

Parkinson t ~a u 

uw - i- ed., Masterworks 
^ of Prose 

ftC0D BUA Mi RANDOLAj 0n the 
dignity of Man, On 
gang and the One, and 

Plato, Epistles 

Euthyphro, Apology, Crito 











Bluck, R., Plato's 

Cornford, F., Plato and 
Plato's Cosmology 
Plato's Theory of 
Hackforth, R., Plato's 
Examination of 
Plato's Phaedo 
Plato's Phaedrus 

Plautus, The Haunted House 
The Menaechmi 
The Rope 

Pope, A., An Essay on Man 

Post, C, ed., Significant Cases in 
British Constitutional 

Quintilian, On the Early 
Education of the 

Reynolds, J., Discourses on Art 

Roman Drama, Copley and Hadas, 

Rosenmeyer, Ostwald, and 

Halporn, The Meters of 
Greek and Latin Poetry 

Russell, B., Philosophy of Science 

Sappho, The Poems of 

Schiller, J., Wilhelm Tell 
Schlegel, J., On Imitation and 

Other Essays 
Schneider, H., Sources of 


Philosophical Realism 

in America 
Schopenhauer, A., On the Basis 

of Morality 
Freedom of the Will 

Selby-Bigge, L., British Moralists 
Seneca, Medea 



Shaftesbury, A., Characteristics 
Shelley, P., A Defence of Poetry 
Smith, A., The Wealth of Nations 

Song of Roland, Terry, trans. 
Sophocles, Electra 

Spiegelberg, H., The Socratic 

Spinoza, B., Earlier Philosophical 
On the Improvement of the 
Terence, The Brothers 
The Eunuch 
The Mother-in-Law 

The Self-Tormentor 
The Woman of Andros 
Three Greek Romances, Hadas, 

Tolstoy, L., What is Art? 
Vergil, Aeneid 

Vico, G. B., On the Study Methods 

Our Time 
Voltaire, Philosophical Letters 
Whitehead, A., Interpretation of 


Wolff, C, Preliminary Discourse 
on Philosophy in General 

Xenophon, Recollections of 

Socrates and Socrates' 
Defense Before the Jury